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<div class="section" id="gpy-likelihoods-package">
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<h1>GPy.likelihoods package<a class="headerlink" href="#gpy-likelihoods-package" title="Permalink to this headline">¶</a></h1>
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<div class="section" id="submodules">
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<h2>Submodules<a class="headerlink" href="#submodules" title="Permalink to this headline">¶</a></h2>
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</div>
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<div class="section" id="module-GPy.likelihoods.bernoulli">
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<span id="gpy-likelihoods-bernoulli-module"></span><h2>GPy.likelihoods.bernoulli module<a class="headerlink" href="#module-GPy.likelihoods.bernoulli" title="Permalink to this headline">¶</a></h2>
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<dl class="class">
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<dt id="GPy.likelihoods.bernoulli.Bernoulli">
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<em class="property">class </em><code class="descclassname">GPy.likelihoods.bernoulli.</code><code class="descname">Bernoulli</code><span class="sig-paren">(</span><em>gp_link=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/bernoulli.html#Bernoulli"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.bernoulli.Bernoulli" title="Permalink to this definition">¶</a></dt>
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<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.likelihood.Likelihood" title="GPy.likelihoods.likelihood.Likelihood"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.likelihood.Likelihood</span></code></a></p>
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<p>Bernoulli likelihood</p>
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<div class="math">
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<p><img src="_images/math/ae9f0580cb8ea29c74c8e96c3d9d7d4d77ba3e46.png" alt="p(y_{i}|\lambda(f_{i})) = \lambda(f_{i})^{y_{i}}(1-f_{i})^{1-y_{i}}"/></p>
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</div><div class="admonition note">
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<p class="first admonition-title">Note</p>
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<p class="last">Y takes values in either {-1, 1} or {0, 1}.
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link function should have the domain [0, 1], e.g. probit (default) or Heaviside</p>
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</div>
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<dl class="method">
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<dt id="GPy.likelihoods.bernoulli.Bernoulli.d2logpdf_dlink2">
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<code class="descname">d2logpdf_dlink2</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/bernoulli.html#Bernoulli.d2logpdf_dlink2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.bernoulli.Bernoulli.d2logpdf_dlink2" title="Permalink to this definition">¶</a></dt>
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<dd><p>Hessian at y, given inv_link_f, w.r.t inv_link_f the hessian will be 0 unless i == j
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i.e. second derivative logpdf at y given inverse link of f_i and inverse link of f_j w.r.t inverse link of f_i and inverse link of f_j.</p>
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<div class="math">
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<p><img src="_images/math/f9939fafeac3df2cd993cd6d46762f5c9e184253.png" alt="\frac{d^{2}\ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)^{2}} = \frac{-y_{i}}{\lambda(f)^{2}} - \frac{(1-y_{i})}{(1-\lambda(f))^{2}}"/></p>
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</div><table class="docutils field-list" frame="void" rules="none">
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<col class="field-name" />
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<col class="field-body" />
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<tbody valign="top">
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<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
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<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables inverse link of f.</li>
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<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
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<li><strong>Y_metadata</strong> – Y_metadata not used in bernoulli</li>
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</ul>
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</td>
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</tr>
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<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">Diagonal of log hessian matrix (second derivative of log likelihood evaluated at points inverse link of f.</p>
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</td>
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</tr>
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<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
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</td>
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</tr>
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</tbody>
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</table>
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<div class="admonition note">
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<p class="first admonition-title">Note</p>
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<p class="last">Will return diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
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(the distribution for y_i depends only on inverse link of f_i not on inverse link of f_(j!=i)</p>
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</div>
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</dd></dl>
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<dl class="method">
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<dt id="GPy.likelihoods.bernoulli.Bernoulli.d3logpdf_dlink3">
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<code class="descname">d3logpdf_dlink3</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/bernoulli.html#Bernoulli.d3logpdf_dlink3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.bernoulli.Bernoulli.d3logpdf_dlink3" title="Permalink to this definition">¶</a></dt>
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<dd><p>Third order derivative log-likelihood function at y given inverse link of f w.r.t inverse link of f</p>
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<div class="math">
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<p><img src="_images/math/61987fe37668cce0640009518e7aa9451c370b03.png" alt="\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = \frac{2y_{i}}{\lambda(f)^{3}} - \frac{2(1-y_{i}}{(1-\lambda(f))^{3}}"/></p>
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</div><table class="docutils field-list" frame="void" rules="none">
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|
<col class="field-name" />
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<col class="field-body" />
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<tbody valign="top">
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<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
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<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables passed through inverse link of f.</li>
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<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
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<li><strong>Y_metadata</strong> – Y_metadata not used in bernoulli</li>
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</ul>
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</td>
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</tr>
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<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">third derivative of log likelihood evaluated at points inverse_link(f)</p>
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</td>
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</tr>
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<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
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</td>
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</tr>
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</tbody>
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</table>
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</dd></dl>
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<dl class="method">
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<dt id="GPy.likelihoods.bernoulli.Bernoulli.dlogpdf_dlink">
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<code class="descname">dlogpdf_dlink</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/bernoulli.html#Bernoulli.dlogpdf_dlink"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.bernoulli.Bernoulli.dlogpdf_dlink" title="Permalink to this definition">¶</a></dt>
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<dd><p>Gradient of the pdf at y, given inverse link of f w.r.t inverse link of f.</p>
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<div class="math">
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<p><img src="_images/math/4cd3f22296f0a10cf5a10a200669957cf04dc550.png" alt="\frac{d\ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \frac{y_{i}}{\lambda(f_{i})} - \frac{(1 - y_{i})}{(1 - \lambda(f_{i}))}"/></p>
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</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables inverse link of f.</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
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<li><strong>Y_metadata</strong> – Y_metadata not used in bernoulli</li>
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</ul>
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</td>
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</tr>
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<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">gradient of log likelihood evaluated at points inverse link of f.</p>
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</td>
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</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
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|
</td>
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|
</tr>
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</tbody>
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</table>
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</dd></dl>
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<dl class="method">
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<dt id="GPy.likelihoods.bernoulli.Bernoulli.exact_inference_gradients">
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<code class="descname">exact_inference_gradients</code><span class="sig-paren">(</span><em>dL_dKdiag</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/bernoulli.html#Bernoulli.exact_inference_gradients"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.bernoulli.Bernoulli.exact_inference_gradients" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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<dl class="method">
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<dt id="GPy.likelihoods.bernoulli.Bernoulli.logpdf_link">
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<code class="descname">logpdf_link</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/bernoulli.html#Bernoulli.logpdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.bernoulli.Bernoulli.logpdf_link" title="Permalink to this definition">¶</a></dt>
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<dd><p>Log Likelihood function given inverse link of f.</p>
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<div class="math">
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<p><img src="_images/math/6b92b5aded6d4eb0dd8b17739862bf1d347d3ea3.png" alt="\ln p(y_{i}|\lambda(f_{i})) = y_{i}\log\lambda(f_{i}) + (1-y_{i})\log (1-f_{i})"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
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|
<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables inverse link of f.</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
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|
<li><strong>Y_metadata</strong> – Y_metadata not used in bernoulli</li>
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</ul>
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</td>
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|
</tr>
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|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">log likelihood evaluated at points inverse link of f.</p>
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|
</td>
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</tr>
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|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
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</td>
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</tr>
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</tbody>
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|
</table>
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</dd></dl>
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<dl class="method">
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<dt id="GPy.likelihoods.bernoulli.Bernoulli.moments_match_ep">
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<code class="descname">moments_match_ep</code><span class="sig-paren">(</span><em>Y_i</em>, <em>tau_i</em>, <em>v_i</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/bernoulli.html#Bernoulli.moments_match_ep"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.bernoulli.Bernoulli.moments_match_ep" title="Permalink to this definition">¶</a></dt>
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<dd><p>Moments match of the marginal approximation in EP algorithm</p>
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<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
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<li><strong>i</strong> – number of observation (int)</li>
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<li><strong>tau_i</strong> – precision of the cavity distribution (float)</li>
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<li><strong>v_i</strong> – mean/variance of the cavity distribution (float)</li>
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</ul>
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</td>
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</tr>
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</tbody>
|
|
</table>
|
|
</dd></dl>
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<dl class="method">
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<dt id="GPy.likelihoods.bernoulli.Bernoulli.pdf_link">
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<code class="descname">pdf_link</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/bernoulli.html#Bernoulli.pdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.bernoulli.Bernoulli.pdf_link" title="Permalink to this definition">¶</a></dt>
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<dd><p>Likelihood function given inverse link of f.</p>
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<div class="math">
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<p><img src="_images/math/ae9f0580cb8ea29c74c8e96c3d9d7d4d77ba3e46.png" alt="p(y_{i}|\lambda(f_{i})) = \lambda(f_{i})^{y_{i}}(1-f_{i})^{1-y_{i}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables inverse link of f.</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata not used in bernoulli</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
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<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">likelihood evaluated for this point</p>
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</td>
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</tr>
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<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
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|
</td>
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</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
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|
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<dl class="method">
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<dt id="GPy.likelihoods.bernoulli.Bernoulli.predictive_mean">
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<code class="descname">predictive_mean</code><span class="sig-paren">(</span><em>mu</em>, <em>variance</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/bernoulli.html#Bernoulli.predictive_mean"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.bernoulli.Bernoulli.predictive_mean" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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<dl class="method">
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<dt id="GPy.likelihoods.bernoulli.Bernoulli.predictive_variance">
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<code class="descname">predictive_variance</code><span class="sig-paren">(</span><em>mu</em>, <em>variance</em>, <em>pred_mean</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/bernoulli.html#Bernoulli.predictive_variance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.bernoulli.Bernoulli.predictive_variance" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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<dl class="method">
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<dt id="GPy.likelihoods.bernoulli.Bernoulli.samples">
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<code class="descname">samples</code><span class="sig-paren">(</span><em>gp</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/bernoulli.html#Bernoulli.samples"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.bernoulli.Bernoulli.samples" title="Permalink to this definition">¶</a></dt>
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<dd><p>Returns a set of samples of observations based on a given value of the latent variable.</p>
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<table class="docutils field-list" frame="void" rules="none">
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<col class="field-name" />
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<col class="field-body" />
|
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<tbody valign="top">
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<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>gp</strong> – latent variable</td>
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</tr>
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</tbody>
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</table>
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</dd></dl>
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</dd></dl>
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</div>
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<div class="section" id="module-GPy.likelihoods.exponential">
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<span id="gpy-likelihoods-exponential-module"></span><h2>GPy.likelihoods.exponential module<a class="headerlink" href="#module-GPy.likelihoods.exponential" title="Permalink to this headline">¶</a></h2>
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<dl class="class">
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<dt id="GPy.likelihoods.exponential.Exponential">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.exponential.</code><code class="descname">Exponential</code><span class="sig-paren">(</span><em>gp_link=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/exponential.html#Exponential"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.exponential.Exponential" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.likelihood.Likelihood" title="GPy.likelihoods.likelihood.Likelihood"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.likelihood.Likelihood</span></code></a></p>
|
|
<p>Expoential likelihood
|
|
Y is expected to take values in {0,1,2,...}
|
|
—–
|
|
$$
|
|
L(x) = exp(lambda) * lambda**Y_i / Y_i!
|
|
$$</p>
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.exponential.Exponential.d2logpdf_dlink2">
|
|
<code class="descname">d2logpdf_dlink2</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/exponential.html#Exponential.d2logpdf_dlink2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.exponential.Exponential.d2logpdf_dlink2" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Hessian at y, given link(f), w.r.t link(f)
|
|
i.e. second derivative logpdf at y given link(f_i) and link(f_j) w.r.t link(f_i) and link(f_j)
|
|
The hessian will be 0 unless i == j</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/0b03b7b9ecad878eac8336d77abfce94453f7943.png" alt="\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}\lambda(f)} = -\frac{1}{\lambda(f_{i})^{2}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in exponential distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<div class="admonition note">
|
|
<p class="first admonition-title">Note</p>
|
|
<p class="last">Will return diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
|
|
(the distribution for y_i depends only on link(f_i) not on link(f_(j!=i))</p>
|
|
</div>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.exponential.Exponential.d3logpdf_dlink3">
|
|
<code class="descname">d3logpdf_dlink3</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/exponential.html#Exponential.d3logpdf_dlink3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.exponential.Exponential.d3logpdf_dlink3" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Third order derivative log-likelihood function at y given link(f) w.r.t link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/604066808935cbbebeeb48a5590ae334a9511f26.png" alt="\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = \frac{2}{\lambda(f_{i})^{3}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in exponential distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">third derivative of likelihood evaluated at points f</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.exponential.Exponential.dlogpdf_dlink">
|
|
<code class="descname">dlogpdf_dlink</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/exponential.html#Exponential.dlogpdf_dlink"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.exponential.Exponential.dlogpdf_dlink" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Gradient of the log likelihood function at y, given link(f) w.r.t link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/4069809108e9e054f33c34a3a488f6f65caf7c59.png" alt="\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \frac{1}{\lambda(f)} - y_{i}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables (f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in exponential distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">gradient of likelihood evaluated at points</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.exponential.Exponential.logpdf_link">
|
|
<code class="descname">logpdf_link</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/exponential.html#Exponential.logpdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.exponential.Exponential.logpdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Log Likelihood Function given link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/68a2d79b5ad6f38311ea4b1bc63bdbc4b67df02b.png" alt="\ln p(y_{i}|\lambda(f_{i})) = \ln \lambda(f_{i}) - y_{i}\lambda(f_{i})"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables (link(f))</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in exponential distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.exponential.Exponential.pdf_link">
|
|
<code class="descname">pdf_link</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/exponential.html#Exponential.pdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.exponential.Exponential.pdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Likelihood function given link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/72c94abc177c437d0c0be762987594d986bcf711.png" alt="p(y_{i}|\lambda(f_{i})) = \lambda(f_{i})\exp (-y\lambda(f_{i}))"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in exponential distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.exponential.Exponential.samples">
|
|
<code class="descname">samples</code><span class="sig-paren">(</span><em>gp</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/exponential.html#Exponential.samples"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.exponential.Exponential.samples" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Returns a set of samples of observations based on a given value of the latent variable.</p>
|
|
<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>gp</strong> – latent variable</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
</dd></dl>
|
|
|
|
</div>
|
|
<div class="section" id="module-GPy.likelihoods.gamma">
|
|
<span id="gpy-likelihoods-gamma-module"></span><h2>GPy.likelihoods.gamma module<a class="headerlink" href="#module-GPy.likelihoods.gamma" title="Permalink to this headline">¶</a></h2>
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.gamma.Gamma">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.gamma.</code><code class="descname">Gamma</code><span class="sig-paren">(</span><em>gp_link=None</em>, <em>beta=1.0</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gamma.html#Gamma"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gamma.Gamma" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.likelihood.Likelihood" title="GPy.likelihoods.likelihood.Likelihood"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.likelihood.Likelihood</span></code></a></p>
|
|
<p>Gamma likelihood</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/3eccf5d08b634e400733f5568da0e3d0c64383ac.png" alt="p(y_{i}|\lambda(f_{i})) = \frac{\beta^{\alpha_{i}}}{\Gamma(\alpha_{i})}y_{i}^{\alpha_{i}-1}e^{-\beta y_{i}}\\
|
|
\alpha_{i} = \beta y_{i}"/></p>
|
|
</div><dl class="method">
|
|
<dt id="GPy.likelihoods.gamma.Gamma.d2logpdf_dlink2">
|
|
<code class="descname">d2logpdf_dlink2</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gamma.html#Gamma.d2logpdf_dlink2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gamma.Gamma.d2logpdf_dlink2" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Hessian at y, given link(f), w.r.t link(f)
|
|
i.e. second derivative logpdf at y given link(f_i) and link(f_j) w.r.t link(f_i) and link(f_j)
|
|
The hessian will be 0 unless i == j</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/afb7f8aa812a6bafd2d1c4423555e766bce85146.png" alt="\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}\lambda(f)} = -\beta^{2}\frac{d\Psi(\alpha_{i})}{d\alpha_{i}}\\
|
|
\alpha_{i} = \beta y_{i}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in gamma distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<div class="admonition note">
|
|
<p class="first admonition-title">Note</p>
|
|
<p class="last">Will return diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
|
|
(the distribution for y_i depends only on link(f_i) not on link(f_(j!=i))</p>
|
|
</div>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gamma.Gamma.d3logpdf_dlink3">
|
|
<code class="descname">d3logpdf_dlink3</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gamma.html#Gamma.d3logpdf_dlink3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gamma.Gamma.d3logpdf_dlink3" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Third order derivative log-likelihood function at y given link(f) w.r.t link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/0ee9a0a290af1db274c2ea2ef1cae298bc701ce4.png" alt="\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = -\beta^{3}\frac{d^{2}\Psi(\alpha_{i})}{d\alpha_{i}}\\
|
|
\alpha_{i} = \beta y_{i}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in gamma distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">third derivative of likelihood evaluated at points f</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gamma.Gamma.dlogpdf_dlink">
|
|
<code class="descname">dlogpdf_dlink</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gamma.html#Gamma.dlogpdf_dlink"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gamma.Gamma.dlogpdf_dlink" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Gradient of the log likelihood function at y, given link(f) w.r.t link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/78f08794d1aed52b66ce2b784d263d3e7376acf0.png" alt="\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \beta (\log \beta y_{i}) - \Psi(\alpha_{i})\beta\\
|
|
\alpha_{i} = \beta y_{i}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables (f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in gamma distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">gradient of likelihood evaluated at points</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gamma.Gamma.logpdf_link">
|
|
<code class="descname">logpdf_link</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gamma.html#Gamma.logpdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gamma.Gamma.logpdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Log Likelihood Function given link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/42ecdfc10af100dfbbc3ec3b19efda0433ba1580.png" alt="\ln p(y_{i}|\lambda(f_{i})) = \alpha_{i}\log \beta - \log \Gamma(\alpha_{i}) + (\alpha_{i} - 1)\log y_{i} - \beta y_{i}\\
|
|
\alpha_{i} = \beta y_{i}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables (link(f))</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in poisson distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gamma.Gamma.pdf_link">
|
|
<code class="descname">pdf_link</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gamma.html#Gamma.pdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gamma.Gamma.pdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Likelihood function given link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/3eccf5d08b634e400733f5568da0e3d0c64383ac.png" alt="p(y_{i}|\lambda(f_{i})) = \frac{\beta^{\alpha_{i}}}{\Gamma(\alpha_{i})}y_{i}^{\alpha_{i}-1}e^{-\beta y_{i}}\\
|
|
\alpha_{i} = \beta y_{i}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in poisson distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
</dd></dl>
|
|
|
|
</div>
|
|
<div class="section" id="module-GPy.likelihoods.gaussian">
|
|
<span id="gpy-likelihoods-gaussian-module"></span><h2>GPy.likelihoods.gaussian module<a class="headerlink" href="#module-GPy.likelihoods.gaussian" title="Permalink to this headline">¶</a></h2>
|
|
<p>A lot of this code assumes that the link function is the identity.</p>
|
|
<p>I think laplace code is okay, but I’m quite sure that the EP moments will only work if the link is identity.</p>
|
|
<p>Furthermore, exact Guassian inference can only be done for the identity link, so we should be asserting so for all calls which relate to that.</p>
|
|
<p>James 11/12/13</p>
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.gaussian.</code><code class="descname">Gaussian</code><span class="sig-paren">(</span><em>gp_link=None</em>, <em>variance=1.0</em>, <em>name='Gaussian_noise'</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.likelihood.Likelihood" title="GPy.likelihoods.likelihood.Likelihood"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.likelihood.Likelihood</span></code></a></p>
|
|
<p>Gaussian likelihood</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/e67edbe203d60c70e432522604959c63318ca2c4.png" alt="\ln p(y_{i}|\lambda(f_{i})) = -\frac{N \ln 2\pi}{2} - \frac{\ln |K|}{2} - \frac{(y_{i} - \lambda(f_{i}))^{T}\sigma^{-2}(y_{i} - \lambda(f_{i}))}{2}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
|
|
<li><strong>variance</strong> – variance value of the Gaussian distribution</li>
|
|
<li><strong>N</strong> (<em>int</em>) – Number of data points</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.betaY">
|
|
<code class="descname">betaY</code><span class="sig-paren">(</span><em>Y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.betaY"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.betaY" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.d2logpdf_dlink2">
|
|
<code class="descname">d2logpdf_dlink2</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.d2logpdf_dlink2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.d2logpdf_dlink2" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Hessian at y, given link_f, w.r.t link_f.
|
|
i.e. second derivative logpdf at y given link(f_i) link(f_j) w.r.t link(f_i) and link(f_j)</p>
|
|
<p>The hessian will be 0 unless i == j</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/143f49a801c035369b4e0e7ca765346063e4686d.png" alt="\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}f} = -\frac{1}{\sigma^{2}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata not used in gaussian</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">Diagonal of log hessian matrix (second derivative of log likelihood evaluated at points link(f))</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<div class="admonition note">
|
|
<p class="first admonition-title">Note</p>
|
|
<p class="last">Will return diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
|
|
(the distribution for y_i depends only on link(f_i) not on link(f_(j!=i))</p>
|
|
</div>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.d2logpdf_dlink2_dtheta">
|
|
<code class="descname">d2logpdf_dlink2_dtheta</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.d2logpdf_dlink2_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.d2logpdf_dlink2_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.d2logpdf_dlink2_dvar">
|
|
<code class="descname">d2logpdf_dlink2_dvar</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.d2logpdf_dlink2_dvar"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.d2logpdf_dlink2_dvar" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Gradient of the hessian (d2logpdf_dlink2) w.r.t variance parameter (noise_variance)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/908550a4d7fbf593ce9180499b72adbf17b50d99.png" alt="\frac{d}{d\sigma^{2}}(\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}\lambda(f)}) = \frac{1}{\sigma^{4}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata not used in gaussian</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">derivative of log hessian evaluated at points link(f_i) and link(f_j) w.r.t variance parameter</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.d3logpdf_dlink3">
|
|
<code class="descname">d3logpdf_dlink3</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.d3logpdf_dlink3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.d3logpdf_dlink3" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Third order derivative log-likelihood function at y given link(f) w.r.t link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/dd6e77999bb2b0a71f5089b6a48aabd729045633.png" alt="\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = 0"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata not used in gaussian</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">third derivative of log likelihood evaluated at points link(f)</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.dlogpdf_dlink">
|
|
<code class="descname">dlogpdf_dlink</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.dlogpdf_dlink"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.dlogpdf_dlink" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Gradient of the pdf at y, given link(f) w.r.t link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/afe71d9f26a15da7a8f28a7b3a66fd7f477b5df7.png" alt="\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \frac{1}{\sigma^{2}}(y_{i} - \lambda(f_{i}))"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata not used in gaussian</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">gradient of log likelihood evaluated at points link(f)</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.dlogpdf_dlink_dtheta">
|
|
<code class="descname">dlogpdf_dlink_dtheta</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.dlogpdf_dlink_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.dlogpdf_dlink_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.dlogpdf_dlink_dvar">
|
|
<code class="descname">dlogpdf_dlink_dvar</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.dlogpdf_dlink_dvar"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.dlogpdf_dlink_dvar" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Derivative of the dlogpdf_dlink w.r.t variance parameter (noise_variance)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/d46f1c279ce0c199db52748ba5f2feb62a46363b.png" alt="\frac{d}{d\sigma^{2}}(\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)}) = \frac{1}{\sigma^{4}}(-y_{i} + \lambda(f_{i}))"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata not used in gaussian</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">derivative of log likelihood evaluated at points link(f) w.r.t variance parameter</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.dlogpdf_link_dtheta">
|
|
<code class="descname">dlogpdf_link_dtheta</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.dlogpdf_link_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.dlogpdf_link_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.dlogpdf_link_dvar">
|
|
<code class="descname">dlogpdf_link_dvar</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.dlogpdf_link_dvar"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.dlogpdf_link_dvar" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Gradient of the log-likelihood function at y given link(f), w.r.t variance parameter (noise_variance)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/72ccd2a09fe1d38b26319c571d7072d31e08bce3.png" alt="\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\sigma^{2}} = -\frac{N}{2\sigma^{2}} + \frac{(y_{i} - \lambda(f_{i}))^{2}}{2\sigma^{4}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata not used in gaussian</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">derivative of log likelihood evaluated at points link(f) w.r.t variance parameter</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.exact_inference_gradients">
|
|
<code class="descname">exact_inference_gradients</code><span class="sig-paren">(</span><em>dL_dKdiag</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.exact_inference_gradients"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.exact_inference_gradients" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.gaussian_variance">
|
|
<code class="descname">gaussian_variance</code><span class="sig-paren">(</span><em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.gaussian_variance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.gaussian_variance" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.log_predictive_density">
|
|
<code class="descname">log_predictive_density</code><span class="sig-paren">(</span><em>y_test</em>, <em>mu_star</em>, <em>var_star</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.log_predictive_density"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.log_predictive_density" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>assumes independence</p>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.logpdf_link">
|
|
<code class="descname">logpdf_link</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.logpdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.logpdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Log likelihood function given link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/e67edbe203d60c70e432522604959c63318ca2c4.png" alt="\ln p(y_{i}|\lambda(f_{i})) = -\frac{N \ln 2\pi}{2} - \frac{\ln |K|}{2} - \frac{(y_{i} - \lambda(f_{i}))^{T}\sigma^{-2}(y_{i} - \lambda(f_{i}))}{2}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata not used in gaussian</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">log likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.pdf_link">
|
|
<code class="descname">pdf_link</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.pdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.pdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Likelihood function given link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/e67edbe203d60c70e432522604959c63318ca2c4.png" alt="\ln p(y_{i}|\lambda(f_{i})) = -\frac{N \ln 2\pi}{2} - \frac{\ln |K|}{2} - \frac{(y_{i} - \lambda(f_{i}))^{T}\sigma^{-2}(y_{i} - \lambda(f_{i}))}{2}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata not used in gaussian</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.predictive_mean">
|
|
<code class="descname">predictive_mean</code><span class="sig-paren">(</span><em>mu</em>, <em>sigma</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.predictive_mean"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.predictive_mean" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.predictive_quantiles">
|
|
<code class="descname">predictive_quantiles</code><span class="sig-paren">(</span><em>mu</em>, <em>var</em>, <em>quantiles</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.predictive_quantiles"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.predictive_quantiles" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.predictive_values">
|
|
<code class="descname">predictive_values</code><span class="sig-paren">(</span><em>mu</em>, <em>var</em>, <em>full_cov=False</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.predictive_values"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.predictive_values" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.predictive_variance">
|
|
<code class="descname">predictive_variance</code><span class="sig-paren">(</span><em>mu</em>, <em>sigma</em>, <em>predictive_mean=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.predictive_variance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.predictive_variance" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.samples">
|
|
<code class="descname">samples</code><span class="sig-paren">(</span><em>gp</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.samples"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.samples" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Returns a set of samples of observations based on a given value of the latent variable.</p>
|
|
<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>gp</strong> – latent variable</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.gaussian.Gaussian.update_gradients">
|
|
<code class="descname">update_gradients</code><span class="sig-paren">(</span><em>grad</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/gaussian.html#Gaussian.update_gradients"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.gaussian.Gaussian.update_gradients" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
</dd></dl>
|
|
|
|
</div>
|
|
<div class="section" id="module-GPy.likelihoods.likelihood">
|
|
<span id="gpy-likelihoods-likelihood-module"></span><h2>GPy.likelihoods.likelihood module<a class="headerlink" href="#module-GPy.likelihoods.likelihood" title="Permalink to this headline">¶</a></h2>
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.likelihood.</code><code class="descname">Likelihood</code><span class="sig-paren">(</span><em>gp_link</em>, <em>name</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="GPy.core.parameterization.html#GPy.core.parameterization.parameterized.Parameterized" title="GPy.core.parameterization.parameterized.Parameterized"><code class="xref py py-class docutils literal"><span class="pre">GPy.core.parameterization.parameterized.Parameterized</span></code></a></p>
|
|
<p>Likelihood base class, used to defing p(y|f).</p>
|
|
<p>All instances use _inverse_ link functions, which can be swapped out. It is
|
|
expected that inheriting classes define a default inverse link function</p>
|
|
<p>To use this class, inherit and define missing functionality.</p>
|
|
<dl class="docutils">
|
|
<dt>Inheriting classes <em>must</em> implement:</dt>
|
|
<dd>pdf_link : a bound method which turns the output of the link function into the pdf
|
|
logpdf_link : the logarithm of the above</dd>
|
|
<dt>To enable use with EP, inheriting classes <em>must</em> define:</dt>
|
|
<dd>TODO: a suitable derivative function for any parameters of the class</dd>
|
|
<dt>It is also desirable to define:</dt>
|
|
<dd>moments_match_ep : a function to compute the EP moments If this isn’t defined, the moments will be computed using 1D quadrature.</dd>
|
|
<dt>To enable use with Laplace approximation, inheriting classes <em>must</em> define:</dt>
|
|
<dd>Some derivative functions <em>AS TODO</em></dd>
|
|
</dl>
|
|
<p>For exact Gaussian inference, define <em>JH TODO</em></p>
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.conditional_mean">
|
|
<code class="descname">conditional_mean</code><span class="sig-paren">(</span><em>gp</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.conditional_mean"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.conditional_mean" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>The mean of the random variable conditioned on one value of the GP</p>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.conditional_variance">
|
|
<code class="descname">conditional_variance</code><span class="sig-paren">(</span><em>gp</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.conditional_variance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.conditional_variance" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>The variance of the random variable conditioned on one value of the GP</p>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.d2logpdf_df2">
|
|
<code class="descname">d2logpdf_df2</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.d2logpdf_df2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.d2logpdf_df2" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Evaluates the link function link(f) then computes the second derivative of log likelihood using it
|
|
Uses the Faa di Bruno’s formula for the chain rule</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/b75a7991cbb1a7fc99730d8856ca842ee8171eae.png" alt="\frac{d^{2}\log p(y|\lambda(f))}{df^{2}} = \frac{d^{2}\log p(y|\lambda(f))}{d^{2}\lambda(f)}\left(\frac{d\lambda(f)}{df}\right)^{2} + \frac{d\log p(y|\lambda(f))}{d\lambda(f)}\frac{d^{2}\lambda(f)}{df^{2}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>f</strong> (<em>Nx1 array</em>) – latent variables f</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution - not used</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">second derivative of log likelihood evaluated for this point (diagonal only)</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">1xN array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.d2logpdf_df2_dtheta">
|
|
<code class="descname">d2logpdf_df2_dtheta</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.d2logpdf_df2_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.d2logpdf_df2_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>TODO: Doc strings</p>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.d2logpdf_dlink2">
|
|
<code class="descname">d2logpdf_dlink2</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.d2logpdf_dlink2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.d2logpdf_dlink2" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.d2logpdf_dlink2_dtheta">
|
|
<code class="descname">d2logpdf_dlink2_dtheta</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.d2logpdf_dlink2_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.d2logpdf_dlink2_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.d3logpdf_df3">
|
|
<code class="descname">d3logpdf_df3</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.d3logpdf_df3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.d3logpdf_df3" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Evaluates the link function link(f) then computes the third derivative of log likelihood using it
|
|
Uses the Faa di Bruno’s formula for the chain rule</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/d891af7e39e0ff3708b97ef66c7e246a93aee52e.png" alt="\frac{d^{3}\log p(y|\lambda(f))}{df^{3}} = \frac{d^{3}\log p(y|\lambda(f)}{d\lambda(f)^{3}}\left(\frac{d\lambda(f)}{df}\right)^{3} + 3\frac{d^{2}\log p(y|\lambda(f)}{d\lambda(f)^{2}}\frac{d\lambda(f)}{df}\frac{d^{2}\lambda(f)}{df^{2}} + \frac{d\log p(y|\lambda(f)}{d\lambda(f)}\frac{d^{3}\lambda(f)}{df^{3}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>f</strong> (<em>Nx1 array</em>) – latent variables f</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution - not used</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">third derivative of log likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.d3logpdf_dlink3">
|
|
<code class="descname">d3logpdf_dlink3</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.d3logpdf_dlink3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.d3logpdf_dlink3" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.dlogpdf_df">
|
|
<code class="descname">dlogpdf_df</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.dlogpdf_df"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.dlogpdf_df" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Evaluates the link function link(f) then computes the derivative of log likelihood using it
|
|
Uses the Faa di Bruno’s formula for the chain rule</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/a51d2cde832e2fe0f2df87be7c62ffc60a456418.png" alt="\frac{d\log p(y|\lambda(f))}{df} = \frac{d\log p(y|\lambda(f))}{d\lambda(f)}\frac{d\lambda(f)}{df}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>f</strong> (<em>Nx1 array</em>) – latent variables f</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution - not used</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">derivative of log likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">1xN array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.dlogpdf_df_dtheta">
|
|
<code class="descname">dlogpdf_df_dtheta</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.dlogpdf_df_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.dlogpdf_df_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>TODO: Doc strings</p>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.dlogpdf_dlink">
|
|
<code class="descname">dlogpdf_dlink</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.dlogpdf_dlink"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.dlogpdf_dlink" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.dlogpdf_dlink_dtheta">
|
|
<code class="descname">dlogpdf_dlink_dtheta</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.dlogpdf_dlink_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.dlogpdf_dlink_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.dlogpdf_dtheta">
|
|
<code class="descname">dlogpdf_dtheta</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.dlogpdf_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.dlogpdf_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>TODO: Doc strings</p>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.dlogpdf_link_dtheta">
|
|
<code class="descname">dlogpdf_link_dtheta</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.dlogpdf_link_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.dlogpdf_link_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.log_predictive_density">
|
|
<code class="descname">log_predictive_density</code><span class="sig-paren">(</span><em>y_test</em>, <em>mu_star</em>, <em>var_star</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.log_predictive_density"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.log_predictive_density" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Calculation of the log predictive density</p>
|
|
<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
|
|
<li><strong>y_test</strong> (<em>(Nx1) array</em>) – test observations (y_{*})</li>
|
|
<li><strong>mu_star</strong> (<em>(Nx1) array</em>) – predictive mean of gaussian p(f_{*}|mu_{*}, var_{*})</li>
|
|
<li><strong>var_star</strong> (<em>(Nx1) array</em>) – predictive variance of gaussian p(f_{*}|mu_{*}, var_{*})</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.logpdf">
|
|
<code class="descname">logpdf</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.logpdf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.logpdf" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Evaluates the link function link(f) then computes the log likelihood (log pdf) using it</p>
|
|
<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>f</strong> (<em>Nx1 array</em>) – latent variables f</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution - not used</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">log likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.logpdf_link">
|
|
<code class="descname">logpdf_link</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.logpdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.logpdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.pdf">
|
|
<code class="descname">pdf</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.pdf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.pdf" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Evaluates the link function link(f) then computes the likelihood (pdf) using it</p>
|
|
<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>f</strong> (<em>Nx1 array</em>) – latent variables f</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution - not used</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.pdf_link">
|
|
<code class="descname">pdf_link</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.pdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.pdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.predictive_mean">
|
|
<code class="descname">predictive_mean</code><span class="sig-paren">(</span><em>mu</em>, <em>variance</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.predictive_mean"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.predictive_mean" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Quadrature calculation of the predictive mean: E(Y_star|Y) = E( E(Y_star|f_star, Y) )</p>
|
|
<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
|
|
<li><strong>mu</strong> – mean of posterior</li>
|
|
<li><strong>sigma</strong> – standard deviation of posterior</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.predictive_quantiles">
|
|
<code class="descname">predictive_quantiles</code><span class="sig-paren">(</span><em>mu</em>, <em>var</em>, <em>quantiles</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.predictive_quantiles"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.predictive_quantiles" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.predictive_values">
|
|
<code class="descname">predictive_values</code><span class="sig-paren">(</span><em>mu</em>, <em>var</em>, <em>full_cov=False</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.predictive_values"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.predictive_values" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Compute mean, variance of the predictive distibution.</p>
|
|
<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first last simple">
|
|
<li><strong>mu</strong> – mean of the latent variable, f, of posterior</li>
|
|
<li><strong>var</strong> – variance of the latent variable, f, of posterior</li>
|
|
<li><strong>full_cov</strong> (<em>Boolean</em>) – whether to use the full covariance or just the diagonal</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.predictive_variance">
|
|
<code class="descname">predictive_variance</code><span class="sig-paren">(</span><em>mu</em>, <em>variance</em>, <em>predictive_mean=None</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.predictive_variance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.predictive_variance" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Approximation to the predictive variance: V(Y_star)</p>
|
|
<p>The following variance decomposition is used:
|
|
V(Y_star) = E( V(Y_star|f_star) ) + V( E(Y_star|f_star) )</p>
|
|
<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>mu</strong> – mean of posterior</li>
|
|
<li><strong>sigma</strong> – standard deviation of posterior</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name" colspan="2">Predictive_mean:</th></tr>
|
|
<tr class="field-even field"><td> </td><td class="field-body"><p class="first last">output’s predictive mean, if None _predictive_mean function will be called.</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.samples">
|
|
<code class="descname">samples</code><span class="sig-paren">(</span><em>gp</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.samples"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.samples" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Returns a set of samples of observations based on a given value of the latent variable.</p>
|
|
<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>gp</strong> – latent variable</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.update_gradients">
|
|
<code class="descname">update_gradients</code><span class="sig-paren">(</span><em>partial</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.update_gradients"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.update_gradients" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.likelihood.Likelihood.variational_expectations">
|
|
<code class="descname">variational_expectations</code><span class="sig-paren">(</span><em>Y</em>, <em>m</em>, <em>v</em>, <em>gh_points=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/likelihood.html#Likelihood.variational_expectations"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.likelihood.Likelihood.variational_expectations" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Use Gauss-Hermite Quadrature to compute</p>
|
|
<blockquote>
|
|
<div>E_p(f) [ log p(y|f) ]
|
|
d/dm E_p(f) [ log p(y|f) ]
|
|
d/dv E_p(f) [ log p(y|f) ]</div></blockquote>
|
|
<p>where p(f) is a Gaussian with mean m and variance v. The shapes of Y, m and v should match.</p>
|
|
<p>if no gh_points are passed, we construct them using defualt options</p>
|
|
</dd></dl>
|
|
|
|
</dd></dl>
|
|
|
|
</div>
|
|
<div class="section" id="module-GPy.likelihoods.link_functions">
|
|
<span id="gpy-likelihoods-link-functions-module"></span><h2>GPy.likelihoods.link_functions module<a class="headerlink" href="#module-GPy.likelihoods.link_functions" title="Permalink to this headline">¶</a></h2>
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.link_functions.Cloglog">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.link_functions.</code><code class="descname">Cloglog</code><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Cloglog"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Cloglog" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.link_functions.GPTransformation" title="GPy.likelihoods.link_functions.GPTransformation"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.link_functions.GPTransformation</span></code></a></p>
|
|
<p>Complementary log-log link
|
|
.. math:</p>
|
|
<div class="highlight-python"><div class="highlight"><pre>p(f) = 1 - e^{-e^f}
|
|
|
|
or
|
|
|
|
f = \log (-\log(1-p))
|
|
</pre></div>
|
|
</div>
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Cloglog.d2transf_df2">
|
|
<code class="descname">d2transf_df2</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Cloglog.d2transf_df2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Cloglog.d2transf_df2" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Cloglog.d3transf_df3">
|
|
<code class="descname">d3transf_df3</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Cloglog.d3transf_df3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Cloglog.d3transf_df3" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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<dl class="method">
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<dt id="GPy.likelihoods.link_functions.Cloglog.dtransf_df">
|
|
<code class="descname">dtransf_df</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Cloglog.dtransf_df"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Cloglog.dtransf_df" title="Permalink to this definition">¶</a></dt>
|
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<dd></dd></dl>
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<dl class="method">
|
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<dt id="GPy.likelihoods.link_functions.Cloglog.transf">
|
|
<code class="descname">transf</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Cloglog.transf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Cloglog.transf" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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</dd></dl>
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<dl class="class">
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<dt id="GPy.likelihoods.link_functions.GPTransformation">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.link_functions.</code><code class="descname">GPTransformation</code><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#GPTransformation"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.GPTransformation" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <code class="xref py py-class docutils literal"><span class="pre">object</span></code></p>
|
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<p>Link function class for doing non-Gaussian likelihoods approximation</p>
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<table class="docutils field-list" frame="void" rules="none">
|
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<col class="field-name" />
|
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<col class="field-body" />
|
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<tbody valign="top">
|
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<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>Y</strong> – observed output (Nx1 numpy.darray)</td>
|
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</tr>
|
|
</tbody>
|
|
</table>
|
|
<div class="admonition note">
|
|
<p class="first admonition-title">Note</p>
|
|
<p class="last">Y values allowed depend on the likelihood_function used</p>
|
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</div>
|
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<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.GPTransformation.d2transf_df2">
|
|
<code class="descname">d2transf_df2</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#GPTransformation.d2transf_df2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.GPTransformation.d2transf_df2" title="Permalink to this definition">¶</a></dt>
|
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<dd><p>second derivative of transf(f) w.r.t. f</p>
|
|
</dd></dl>
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<dl class="method">
|
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<dt id="GPy.likelihoods.link_functions.GPTransformation.d3transf_df3">
|
|
<code class="descname">d3transf_df3</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#GPTransformation.d3transf_df3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.GPTransformation.d3transf_df3" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>third derivative of transf(f) w.r.t. f</p>
|
|
</dd></dl>
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|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.GPTransformation.dtransf_df">
|
|
<code class="descname">dtransf_df</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#GPTransformation.dtransf_df"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.GPTransformation.dtransf_df" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>derivative of transf(f) w.r.t. f</p>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.GPTransformation.transf">
|
|
<code class="descname">transf</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#GPTransformation.transf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.GPTransformation.transf" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Gaussian process tranformation function, latent space -> output space</p>
|
|
</dd></dl>
|
|
|
|
</dd></dl>
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|
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.link_functions.Heaviside">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.link_functions.</code><code class="descname">Heaviside</code><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Heaviside"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Heaviside" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.link_functions.GPTransformation" title="GPy.likelihoods.link_functions.GPTransformation"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.link_functions.GPTransformation</span></code></a></p>
|
|
<div class="math">
|
|
<p><img src="_images/math/1e2b408025f9da95787112c13a2e7401032a0063.png" alt="g(f) = I_{x \in A}"/></p>
|
|
</div><dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Heaviside.d2transf_df2">
|
|
<code class="descname">d2transf_df2</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Heaviside.d2transf_df2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Heaviside.d2transf_df2" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Heaviside.dtransf_df">
|
|
<code class="descname">dtransf_df</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Heaviside.dtransf_df"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Heaviside.dtransf_df" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Heaviside.transf">
|
|
<code class="descname">transf</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Heaviside.transf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Heaviside.transf" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
</dd></dl>
|
|
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.link_functions.Identity">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.link_functions.</code><code class="descname">Identity</code><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Identity"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Identity" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.link_functions.GPTransformation" title="GPy.likelihoods.link_functions.GPTransformation"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.link_functions.GPTransformation</span></code></a></p>
|
|
<div class="math">
|
|
<p><img src="_images/math/1ee9c98e88e9896c1f62c6486c77b6a83231ccbc.png" alt="g(f) = f"/></p>
|
|
</div><dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Identity.d2transf_df2">
|
|
<code class="descname">d2transf_df2</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Identity.d2transf_df2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Identity.d2transf_df2" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Identity.d3transf_df3">
|
|
<code class="descname">d3transf_df3</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Identity.d3transf_df3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Identity.d3transf_df3" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Identity.dtransf_df">
|
|
<code class="descname">dtransf_df</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Identity.dtransf_df"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Identity.dtransf_df" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Identity.transf">
|
|
<code class="descname">transf</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Identity.transf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Identity.transf" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
</dd></dl>
|
|
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.link_functions.Log">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.link_functions.</code><code class="descname">Log</code><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Log"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Log" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.link_functions.GPTransformation" title="GPy.likelihoods.link_functions.GPTransformation"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.link_functions.GPTransformation</span></code></a></p>
|
|
<div class="math">
|
|
<p><img src="_images/math/30b07cfe9d64b0e6d96edc415ebaaf936b5820ca.png" alt="g(f) = \log(\mu)"/></p>
|
|
</div><dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Log.d2transf_df2">
|
|
<code class="descname">d2transf_df2</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Log.d2transf_df2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Log.d2transf_df2" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Log.d3transf_df3">
|
|
<code class="descname">d3transf_df3</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Log.d3transf_df3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Log.d3transf_df3" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Log.dtransf_df">
|
|
<code class="descname">dtransf_df</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Log.dtransf_df"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Log.dtransf_df" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Log.transf">
|
|
<code class="descname">transf</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Log.transf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Log.transf" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
</dd></dl>
|
|
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.link_functions.Log_ex_1">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.link_functions.</code><code class="descname">Log_ex_1</code><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Log_ex_1"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Log_ex_1" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.link_functions.GPTransformation" title="GPy.likelihoods.link_functions.GPTransformation"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.link_functions.GPTransformation</span></code></a></p>
|
|
<div class="math">
|
|
<p><img src="_images/math/3c943d4fc1aea61edc8246a049f058483725f902.png" alt="g(f) = \log(\exp(\mu) - 1)"/></p>
|
|
</div><dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Log_ex_1.d2transf_df2">
|
|
<code class="descname">d2transf_df2</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Log_ex_1.d2transf_df2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Log_ex_1.d2transf_df2" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Log_ex_1.d3transf_df3">
|
|
<code class="descname">d3transf_df3</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Log_ex_1.d3transf_df3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Log_ex_1.d3transf_df3" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Log_ex_1.dtransf_df">
|
|
<code class="descname">dtransf_df</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Log_ex_1.dtransf_df"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Log_ex_1.dtransf_df" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Log_ex_1.transf">
|
|
<code class="descname">transf</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Log_ex_1.transf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Log_ex_1.transf" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
</dd></dl>
|
|
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.link_functions.Probit">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.link_functions.</code><code class="descname">Probit</code><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Probit"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Probit" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.link_functions.GPTransformation" title="GPy.likelihoods.link_functions.GPTransformation"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.link_functions.GPTransformation</span></code></a></p>
|
|
<div class="math">
|
|
<p><img src="_images/math/93d57b731d01eff6d0e66b973f5d2768e5347a4a.png" alt="g(f) = \Phi^{-1} (mu)"/></p>
|
|
</div><dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Probit.d2transf_df2">
|
|
<code class="descname">d2transf_df2</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Probit.d2transf_df2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Probit.d2transf_df2" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Probit.d3transf_df3">
|
|
<code class="descname">d3transf_df3</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Probit.d3transf_df3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Probit.d3transf_df3" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Probit.dtransf_df">
|
|
<code class="descname">dtransf_df</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Probit.dtransf_df"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Probit.dtransf_df" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Probit.transf">
|
|
<code class="descname">transf</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Probit.transf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Probit.transf" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
</dd></dl>
|
|
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.link_functions.Reciprocal">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.link_functions.</code><code class="descname">Reciprocal</code><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Reciprocal"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Reciprocal" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.link_functions.GPTransformation" title="GPy.likelihoods.link_functions.GPTransformation"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.link_functions.GPTransformation</span></code></a></p>
|
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<dl class="method">
|
|
<dt id="GPy.likelihoods.link_functions.Reciprocal.d2transf_df2">
|
|
<code class="descname">d2transf_df2</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Reciprocal.d2transf_df2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Reciprocal.d2transf_df2" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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<dl class="method">
|
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<dt id="GPy.likelihoods.link_functions.Reciprocal.d3transf_df3">
|
|
<code class="descname">d3transf_df3</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Reciprocal.d3transf_df3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Reciprocal.d3transf_df3" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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<dl class="method">
|
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<dt id="GPy.likelihoods.link_functions.Reciprocal.dtransf_df">
|
|
<code class="descname">dtransf_df</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Reciprocal.dtransf_df"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Reciprocal.dtransf_df" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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<dl class="method">
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<dt id="GPy.likelihoods.link_functions.Reciprocal.transf">
|
|
<code class="descname">transf</code><span class="sig-paren">(</span><em>f</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/link_functions.html#Reciprocal.transf"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.link_functions.Reciprocal.transf" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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</dd></dl>
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</div>
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<div class="section" id="module-GPy.likelihoods.mixed_noise">
|
|
<span id="gpy-likelihoods-mixed-noise-module"></span><h2>GPy.likelihoods.mixed_noise module<a class="headerlink" href="#module-GPy.likelihoods.mixed_noise" title="Permalink to this headline">¶</a></h2>
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<dl class="class">
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<dt id="GPy.likelihoods.mixed_noise.MixedNoise">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.mixed_noise.</code><code class="descname">MixedNoise</code><span class="sig-paren">(</span><em>likelihoods_list</em>, <em>name='mixed_noise'</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/mixed_noise.html#MixedNoise"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.mixed_noise.MixedNoise" title="Permalink to this definition">¶</a></dt>
|
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<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.likelihood.Likelihood" title="GPy.likelihoods.likelihood.Likelihood"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.likelihood.Likelihood</span></code></a></p>
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<dl class="method">
|
|
<dt id="GPy.likelihoods.mixed_noise.MixedNoise.betaY">
|
|
<code class="descname">betaY</code><span class="sig-paren">(</span><em>Y</em>, <em>Y_metadata</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/mixed_noise.html#MixedNoise.betaY"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.mixed_noise.MixedNoise.betaY" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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<dl class="method">
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<dt id="GPy.likelihoods.mixed_noise.MixedNoise.exact_inference_gradients">
|
|
<code class="descname">exact_inference_gradients</code><span class="sig-paren">(</span><em>dL_dKdiag</em>, <em>Y_metadata</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/mixed_noise.html#MixedNoise.exact_inference_gradients"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.mixed_noise.MixedNoise.exact_inference_gradients" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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<dl class="method">
|
|
<dt id="GPy.likelihoods.mixed_noise.MixedNoise.gaussian_variance">
|
|
<code class="descname">gaussian_variance</code><span class="sig-paren">(</span><em>Y_metadata</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/mixed_noise.html#MixedNoise.gaussian_variance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.mixed_noise.MixedNoise.gaussian_variance" title="Permalink to this definition">¶</a></dt>
|
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<dd></dd></dl>
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<dl class="method">
|
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<dt id="GPy.likelihoods.mixed_noise.MixedNoise.predictive_quantiles">
|
|
<code class="descname">predictive_quantiles</code><span class="sig-paren">(</span><em>mu</em>, <em>var</em>, <em>quantiles</em>, <em>Y_metadata</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/mixed_noise.html#MixedNoise.predictive_quantiles"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.mixed_noise.MixedNoise.predictive_quantiles" title="Permalink to this definition">¶</a></dt>
|
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<dd></dd></dl>
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<dl class="method">
|
|
<dt id="GPy.likelihoods.mixed_noise.MixedNoise.predictive_values">
|
|
<code class="descname">predictive_values</code><span class="sig-paren">(</span><em>mu</em>, <em>var</em>, <em>full_cov=False</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/mixed_noise.html#MixedNoise.predictive_values"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.mixed_noise.MixedNoise.predictive_values" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
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|
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<dl class="method">
|
|
<dt id="GPy.likelihoods.mixed_noise.MixedNoise.predictive_variance">
|
|
<code class="descname">predictive_variance</code><span class="sig-paren">(</span><em>mu</em>, <em>sigma</em>, <em>Y_metadata</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/mixed_noise.html#MixedNoise.predictive_variance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.mixed_noise.MixedNoise.predictive_variance" title="Permalink to this definition">¶</a></dt>
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<dd></dd></dl>
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<dl class="method">
|
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<dt id="GPy.likelihoods.mixed_noise.MixedNoise.samples">
|
|
<code class="descname">samples</code><span class="sig-paren">(</span><em>gp</em>, <em>Y_metadata</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/mixed_noise.html#MixedNoise.samples"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.mixed_noise.MixedNoise.samples" title="Permalink to this definition">¶</a></dt>
|
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<dd><p>Returns a set of samples of observations based on a given value of the latent variable.</p>
|
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<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>gp</strong> – latent variable</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.mixed_noise.MixedNoise.update_gradients">
|
|
<code class="descname">update_gradients</code><span class="sig-paren">(</span><em>gradients</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/mixed_noise.html#MixedNoise.update_gradients"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.mixed_noise.MixedNoise.update_gradients" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
</dd></dl>
|
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|
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</div>
|
|
<div class="section" id="module-GPy.likelihoods.poisson">
|
|
<span id="gpy-likelihoods-poisson-module"></span><h2>GPy.likelihoods.poisson module<a class="headerlink" href="#module-GPy.likelihoods.poisson" title="Permalink to this headline">¶</a></h2>
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.poisson.Poisson">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.poisson.</code><code class="descname">Poisson</code><span class="sig-paren">(</span><em>gp_link=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/poisson.html#Poisson"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.poisson.Poisson" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.likelihood.Likelihood" title="GPy.likelihoods.likelihood.Likelihood"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.likelihood.Likelihood</span></code></a></p>
|
|
<p>Poisson likelihood</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/26d17fb3a403d131bb3eef5cfb5c32587282b0d5.png" alt="p(y_{i}|\lambda(f_{i})) = \frac{\lambda(f_{i})^{y_{i}}}{y_{i}!}e^{-\lambda(f_{i})}"/></p>
|
|
</div><div class="admonition note">
|
|
<p class="first admonition-title">Note</p>
|
|
<p class="last">Y is expected to take values in {0,1,2,...}</p>
|
|
</div>
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.poisson.Poisson.conditional_mean">
|
|
<code class="descname">conditional_mean</code><span class="sig-paren">(</span><em>gp</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/poisson.html#Poisson.conditional_mean"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.poisson.Poisson.conditional_mean" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>The mean of the random variable conditioned on one value of the GP</p>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.poisson.Poisson.conditional_variance">
|
|
<code class="descname">conditional_variance</code><span class="sig-paren">(</span><em>gp</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/poisson.html#Poisson.conditional_variance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.poisson.Poisson.conditional_variance" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>The variance of the random variable conditioned on one value of the GP</p>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.poisson.Poisson.d2logpdf_dlink2">
|
|
<code class="descname">d2logpdf_dlink2</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/poisson.html#Poisson.d2logpdf_dlink2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.poisson.Poisson.d2logpdf_dlink2" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Hessian at y, given link(f), w.r.t link(f)
|
|
i.e. second derivative logpdf at y given link(f_i) and link(f_j) w.r.t link(f_i) and link(f_j)
|
|
The hessian will be 0 unless i == j</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/68a7f1a9ea162f83dbb18eb5eeada4da9ec1805d.png" alt="\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}\lambda(f)} = \frac{-y_{i}}{\lambda(f_{i})^{2}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in poisson distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<div class="admonition note">
|
|
<p class="first admonition-title">Note</p>
|
|
<p class="last">Will return diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
|
|
(the distribution for y_i depends only on link(f_i) not on link(f_(j!=i))</p>
|
|
</div>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.poisson.Poisson.d3logpdf_dlink3">
|
|
<code class="descname">d3logpdf_dlink3</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/poisson.html#Poisson.d3logpdf_dlink3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.poisson.Poisson.d3logpdf_dlink3" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Third order derivative log-likelihood function at y given link(f) w.r.t link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/4fbab675639838c2c3f11f9ad372ec258a842318.png" alt="\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = \frac{2y_{i}}{\lambda(f_{i})^{3}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in poisson distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">third derivative of likelihood evaluated at points f</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.poisson.Poisson.dlogpdf_dlink">
|
|
<code class="descname">dlogpdf_dlink</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/poisson.html#Poisson.dlogpdf_dlink"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.poisson.Poisson.dlogpdf_dlink" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Gradient of the log likelihood function at y, given link(f) w.r.t link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/b671e3f640198fb9b743e2e405aad0dfcd645984.png" alt="\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \frac{y_{i}}{\lambda(f_{i})} - 1"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables (f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in poisson distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">gradient of likelihood evaluated at points</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.poisson.Poisson.logpdf_link">
|
|
<code class="descname">logpdf_link</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/poisson.html#Poisson.logpdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.poisson.Poisson.logpdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Log Likelihood Function given link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/f8a932e370ada3f336aca285ab12d4b239473624.png" alt="\ln p(y_{i}|\lambda(f_{i})) = -\lambda(f_{i}) + y_{i}\log \lambda(f_{i}) - \log y_{i}!"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables (link(f))</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in poisson distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.poisson.Poisson.pdf_link">
|
|
<code class="descname">pdf_link</code><span class="sig-paren">(</span><em>link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/poisson.html#Poisson.pdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.poisson.Poisson.pdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Likelihood function given link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/26d17fb3a403d131bb3eef5cfb5c32587282b0d5.png" alt="p(y_{i}|\lambda(f_{i})) = \frac{\lambda(f_{i})^{y_{i}}}{y_{i}!}e^{-\lambda(f_{i})}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in poisson distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.poisson.Poisson.samples">
|
|
<code class="descname">samples</code><span class="sig-paren">(</span><em>gp</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/poisson.html#Poisson.samples"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.poisson.Poisson.samples" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Returns a set of samples of observations based on a given value of the latent variable.</p>
|
|
<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>gp</strong> – latent variable</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
</dd></dl>
|
|
|
|
</div>
|
|
<div class="section" id="module-GPy.likelihoods.student_t">
|
|
<span id="gpy-likelihoods-student-t-module"></span><h2>GPy.likelihoods.student_t module<a class="headerlink" href="#module-GPy.likelihoods.student_t" title="Permalink to this headline">¶</a></h2>
|
|
<dl class="class">
|
|
<dt id="GPy.likelihoods.student_t.StudentT">
|
|
<em class="property">class </em><code class="descclassname">GPy.likelihoods.student_t.</code><code class="descname">StudentT</code><span class="sig-paren">(</span><em>gp_link=None</em>, <em>deg_free=5</em>, <em>sigma2=2</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Bases: <a class="reference internal" href="#GPy.likelihoods.likelihood.Likelihood" title="GPy.likelihoods.likelihood.Likelihood"><code class="xref py py-class docutils literal"><span class="pre">GPy.likelihoods.likelihood.Likelihood</span></code></a></p>
|
|
<p>Student T likelihood</p>
|
|
<p>For nomanclature see Bayesian Data Analysis 2003 p576</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/3a68be8ed48ce0799d2da0027b48dfc94dade8d3.png" alt="p(y_{i}|\lambda(f_{i})) = \frac{\Gamma\left(\frac{v+1}{2}\right)}{\Gamma\left(\frac{v}{2}\right)\sqrt{v\pi\sigma^{2}}}\left(1 + \frac{1}{v}\left(\frac{(y_{i} - f_{i})^{2}}{\sigma^{2}}\right)\right)^{\frac{-v+1}{2}}"/></p>
|
|
</div><dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.conditional_mean">
|
|
<code class="descname">conditional_mean</code><span class="sig-paren">(</span><em>gp</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.conditional_mean"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.conditional_mean" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.conditional_variance">
|
|
<code class="descname">conditional_variance</code><span class="sig-paren">(</span><em>gp</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.conditional_variance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.conditional_variance" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.d2logpdf_dlink2">
|
|
<code class="descname">d2logpdf_dlink2</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.d2logpdf_dlink2"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.d2logpdf_dlink2" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Hessian at y, given link(f), w.r.t link(f)
|
|
i.e. second derivative logpdf at y given link(f_i) and link(f_j) w.r.t link(f_i) and link(f_j)
|
|
The hessian will be 0 unless i == j</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/b9164be40b826a2f164f45571034ed55764d3b7e.png" alt="\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}\lambda(f)} = \frac{(v+1)((y_{i}-\lambda(f_{i}))^{2} - \sigma^{2}v)}{((y_{i}-\lambda(f_{i}))^{2} + \sigma^{2}v)^{2}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables inv_link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
<div class="admonition note">
|
|
<p class="first admonition-title">Note</p>
|
|
<p class="last">Will return diagonal of hessian, since every where else it is 0, as the likelihood factorizes over cases
|
|
(the distribution for y_i depends only on link(f_i) not on link(f_(j!=i))</p>
|
|
</div>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.d2logpdf_dlink2_dtheta">
|
|
<code class="descname">d2logpdf_dlink2_dtheta</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.d2logpdf_dlink2_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.d2logpdf_dlink2_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.d2logpdf_dlink2_dvar">
|
|
<code class="descname">d2logpdf_dlink2_dvar</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.d2logpdf_dlink2_dvar"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.d2logpdf_dlink2_dvar" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Gradient of the hessian (d2logpdf_dlink2) w.r.t variance parameter (t_noise)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/775e11e817ec90bc434b37341dd7c56d71ae492e.png" alt="\frac{d}{d\sigma^{2}}(\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}f}) = \frac{v(v+1)(\sigma^{2}v - 3(y_{i} - \lambda(f_{i}))^{2})}{(\sigma^{2}v + (y_{i} - \lambda(f_{i}))^{2})^{3}}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">derivative of hessian evaluated at points f and f_j w.r.t variance parameter</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.d3logpdf_dlink3">
|
|
<code class="descname">d3logpdf_dlink3</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.d3logpdf_dlink3"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.d3logpdf_dlink3" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Third order derivative log-likelihood function at y given link(f) w.r.t link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/ec9f908da0039cda866af0e06264a6fad9dbed48.png" alt="\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = \frac{-2(v+1)((y_{i} - \lambda(f_{i}))^3 - 3(y_{i} - \lambda(f_{i})) \sigma^{2} v))}{((y_{i} - \lambda(f_{i})) + \sigma^{2} v)^3}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">third derivative of likelihood evaluated at points f</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.dlogpdf_dlink">
|
|
<code class="descname">dlogpdf_dlink</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.dlogpdf_dlink"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.dlogpdf_dlink" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Gradient of the log likelihood function at y, given link(f) w.r.t link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/57f8e051e69685f0bad7e87ab3d5a189b1c713b6.png" alt="\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \frac{(v+1)(y_{i}-\lambda(f_{i}))}{(y_{i}-\lambda(f_{i}))^{2} + \sigma^{2}v}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables (f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">gradient of likelihood evaluated at points</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.dlogpdf_dlink_dtheta">
|
|
<code class="descname">dlogpdf_dlink_dtheta</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.dlogpdf_dlink_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.dlogpdf_dlink_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.dlogpdf_dlink_dvar">
|
|
<code class="descname">dlogpdf_dlink_dvar</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.dlogpdf_dlink_dvar"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.dlogpdf_dlink_dvar" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Derivative of the dlogpdf_dlink w.r.t variance parameter (t_noise)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/4533f6b5623760a1facf22800afd4214b73538b0.png" alt="\frac{d}{d\sigma^{2}}(\frac{d \ln p(y_{i}|\lambda(f_{i}))}{df}) = \frac{-2\sigma v(v + 1)(y_{i}-\lambda(f_{i}))}{(y_{i}-\lambda(f_{i}))^2 + \sigma^2 v)^2}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables inv_link_f</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">derivative of likelihood evaluated at points f w.r.t variance parameter</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">Nx1 array</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.dlogpdf_link_dtheta">
|
|
<code class="descname">dlogpdf_link_dtheta</code><span class="sig-paren">(</span><em>f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.dlogpdf_link_dtheta"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.dlogpdf_link_dtheta" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.dlogpdf_link_dvar">
|
|
<code class="descname">dlogpdf_link_dvar</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.dlogpdf_link_dvar"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.dlogpdf_link_dvar" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Gradient of the log-likelihood function at y given f, w.r.t variance parameter (t_noise)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/f66e06331b1ae3c96e126c035b09ee277f6fc50d.png" alt="\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\sigma^{2}} = \frac{v((y_{i} - \lambda(f_{i}))^{2} - \sigma^{2})}{2\sigma^{2}(\sigma^{2}v + (y_{i} - \lambda(f_{i}))^{2})}"/></p>
|
|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">derivative of likelihood evaluated at points f w.r.t variance parameter</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.logpdf_link">
|
|
<code class="descname">logpdf_link</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.logpdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.logpdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Log Likelihood Function given link(f)</p>
|
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<div class="math">
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|
<p><img src="_images/math/66e9ec52652f5885327c00676d9a4a1c4d4f7f47.png" alt="\ln p(y_{i}|\lambda(f_{i})) = \ln \Gamma\left(\frac{v+1}{2}\right) - \ln \Gamma\left(\frac{v}{2}\right) - \ln \sqrt{v \pi\sigma^{2}} - \frac{v+1}{2}\ln \left(1 + \frac{1}{v}\left(\frac{(y_{i} - \lambda(f_{i}))^{2}}{\sigma^{2}}\right)\right)"/></p>
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</div><table class="docutils field-list" frame="void" rules="none">
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<col class="field-name" />
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<col class="field-body" />
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<tbody valign="top">
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<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables (link(f))</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution</li>
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|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">likelihood evaluated for this point</p>
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|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
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</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
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|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.parameters_changed">
|
|
<code class="descname">parameters_changed</code><span class="sig-paren">(</span><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.parameters_changed"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.parameters_changed" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
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|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.pdf_link">
|
|
<code class="descname">pdf_link</code><span class="sig-paren">(</span><em>inv_link_f</em>, <em>y</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.pdf_link"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.pdf_link" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Likelihood function given link(f)</p>
|
|
<div class="math">
|
|
<p><img src="_images/math/112c6d94605f31bddb3f52529408cf14d2a03456.png" alt="p(y_{i}|\lambda(f_{i})) = \frac{\Gamma\left(\frac{v+1}{2}\right)}{\Gamma\left(\frac{v}{2}\right)\sqrt{v\pi\sigma^{2}}}\left(1 + \frac{1}{v}\left(\frac{(y_{i} - \lambda(f_{i}))^{2}}{\sigma^{2}}\right)\right)^{\frac{-v+1}{2}}"/></p>
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|
</div><table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><ul class="first simple">
|
|
<li><strong>inv_link_f</strong> (<em>Nx1 array</em>) – latent variables link(f)</li>
|
|
<li><strong>y</strong> (<em>Nx1 array</em>) – data</li>
|
|
<li><strong>Y_metadata</strong> – Y_metadata which is not used in student t distribution</li>
|
|
</ul>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-even field"><th class="field-name">Returns:</th><td class="field-body"><p class="first">likelihood evaluated for this point</p>
|
|
</td>
|
|
</tr>
|
|
<tr class="field-odd field"><th class="field-name">Return type:</th><td class="field-body"><p class="first last">float</p>
|
|
</td>
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|
</tr>
|
|
</tbody>
|
|
</table>
|
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</dd></dl>
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|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.predictive_mean">
|
|
<code class="descname">predictive_mean</code><span class="sig-paren">(</span><em>mu</em>, <em>sigma</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.predictive_mean"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.predictive_mean" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.predictive_variance">
|
|
<code class="descname">predictive_variance</code><span class="sig-paren">(</span><em>mu</em>, <em>variance</em>, <em>predictive_mean=None</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.predictive_variance"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.predictive_variance" title="Permalink to this definition">¶</a></dt>
|
|
<dd></dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.samples">
|
|
<code class="descname">samples</code><span class="sig-paren">(</span><em>gp</em>, <em>Y_metadata=None</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.samples"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.samples" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Returns a set of samples of observations based on a given value of the latent variable.</p>
|
|
<table class="docutils field-list" frame="void" rules="none">
|
|
<col class="field-name" />
|
|
<col class="field-body" />
|
|
<tbody valign="top">
|
|
<tr class="field-odd field"><th class="field-name">Parameters:</th><td class="field-body"><strong>gp</strong> – latent variable</td>
|
|
</tr>
|
|
</tbody>
|
|
</table>
|
|
</dd></dl>
|
|
|
|
<dl class="method">
|
|
<dt id="GPy.likelihoods.student_t.StudentT.update_gradients">
|
|
<code class="descname">update_gradients</code><span class="sig-paren">(</span><em>grads</em><span class="sig-paren">)</span><a class="reference internal" href="_modules/GPy/likelihoods/student_t.html#StudentT.update_gradients"><span class="viewcode-link">[source]</span></a><a class="headerlink" href="#GPy.likelihoods.student_t.StudentT.update_gradients" title="Permalink to this definition">¶</a></dt>
|
|
<dd><p>Pull out the gradients, be careful as the order must match the order
|
|
in which the parameters are added</p>
|
|
</dd></dl>
|
|
|
|
</dd></dl>
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|
|
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</div>
|
|
<div class="section" id="module-GPy.likelihoods">
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|
<span id="module-contents"></span><h2>Module contents<a class="headerlink" href="#module-GPy.likelihoods" title="Permalink to this headline">¶</a></h2>
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</div>
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<h3><a href="index.html">Table Of Contents</a></h3>
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<ul>
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<li><a class="reference internal" href="#">GPy.likelihoods package</a><ul>
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<li><a class="reference internal" href="#submodules">Submodules</a></li>
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<li><a class="reference internal" href="#module-GPy.likelihoods.bernoulli">GPy.likelihoods.bernoulli module</a></li>
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<li><a class="reference internal" href="#module-GPy.likelihoods.exponential">GPy.likelihoods.exponential module</a></li>
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<li><a class="reference internal" href="#module-GPy.likelihoods.gamma">GPy.likelihoods.gamma module</a></li>
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<li><a class="reference internal" href="#module-GPy.likelihoods.gaussian">GPy.likelihoods.gaussian module</a></li>
|
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<li><a class="reference internal" href="#module-GPy.likelihoods.likelihood">GPy.likelihoods.likelihood module</a></li>
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<li><a class="reference internal" href="#module-GPy.likelihoods.link_functions">GPy.likelihoods.link_functions module</a></li>
|
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<li><a class="reference internal" href="#module-GPy.likelihoods.mixed_noise">GPy.likelihoods.mixed_noise module</a></li>
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<li><a class="reference internal" href="#module-GPy.likelihoods.poisson">GPy.likelihoods.poisson module</a></li>
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<li><a class="reference internal" href="#module-GPy.likelihoods.student_t">GPy.likelihoods.student_t module</a></li>
|
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<li><a class="reference internal" href="#module-GPy.likelihoods">Module contents</a></li>
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</ul>
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