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Started on chaining, must remember to chain _laplace_gradients aswell!
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Alan Saul
parent
a0aac76812
commit
96f189113a
4 changed files with 325 additions and 235 deletions
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@ -40,64 +40,82 @@ class StudentT(NoiseDistribution):
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def variance(self, extra_data=None):
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return (self.v / float(self.v - 2)) * self.sigma2
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def _nlog_mass(self, gp, obs, extra_data=None):
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def _nlog_mass(self, link_f, y, extra_data=None):
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NotImplementedError("Deprecated, now doing chain in likelihood.py for link function evaluation\
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Please negate your function and use logpdf in noise_model.py, if implementing a likelihood\
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rederivate the derivative without doing the chain and put in logpdf, dlogpdf_dlink or\
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its derivatives")
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def _dnlog_mass_dgp(self, link_f, y, extra_data=None):
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NotImplementedError("Deprecated, now doing chain in likelihood.py for link function evaluation\
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Please negate your function and use logpdf in noise_model.py, if implementing a likelihood\
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rederivate the derivative without doing the chain and put in logpdf, dlogpdf_dlink or\
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its derivatives")
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def _d2nlog_mass_dgp2(self, link_f, y, extra_data=None):
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NotImplementedError("Deprecated, now doing chain in likelihood.py for link function evaluation\
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Please negate your function and use logpdf in noise_model.py, if implementing a likelihood\
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rederivate the derivative without doing the chain and put in logpdf, dlogpdf_dlink or\
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its derivatives")
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def logpdf(self, link_f, y, extra_data=None):
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"""
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Log Likelihood Function
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.. math::
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\\ln p(y_{i}|f_{i}) = \\ln \\Gamma(\\frac{v+1}{2}) - \\ln \\Gamma(\\frac{v}{2})\\sqrt{v \\pi}\sigma - \\frac{v+1}{2}\\ln (1 + \\frac{1}{v}\\left(\\frac{y_{i} - f_{i}}{\\sigma}\\right)^2
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:param gp: latent variables (f)
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:type gp: Nx1 array
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:param obs: data (y)
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:type obs: Nx1 array
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:param link_f: latent variables (link(f))
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:type link_f: Nx1 array
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:param y: data
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:type y: Nx1 array
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:param extra_data: extra_data which is not used in student t distribution - not used
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:returns: likelihood evaluated for this point
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:rtype: float
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"""
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assert gp.shape == obs.shape
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e = obs - self.gp_link.transf(gp)
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assert link_f.shape == y.shape
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e = y - link_f
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objective = (+ gammaln((self.v + 1) * 0.5)
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- gammaln(self.v * 0.5)
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- 0.5*np.log(self.sigma2 * self.v * np.pi)
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- 0.5*(self.v + 1)*np.log(1 + (1/np.float(self.v))*((e**2)/self.sigma2))
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)
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return -np.sum(objective)
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return np.sum(objective)
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def dlik_df(self, y, f, extra_data=None):
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def dlogpdf_dlink(self, link_f, y, extra_data=None):
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"""
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Gradient of the log likelihood function at y, given f w.r.t f
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Gradient of the log likelihood function at y, given link(f) w.r.t link(f)
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.. math::
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\\frac{d \\ln p(y_{i}|f_{i})}{df} = \\frac{(v+1)(y_{i}-f_{i})}{(y_{i}-f_{i})^{2} + \\sigma^{2}v}
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:param link_f: latent variables (f)
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:type link_f: Nx1 array
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:param y: data
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:type y: Nx1 array
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:param f: latent variables f
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:type f: Nx1 array
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:param extra_data: extra_data which is not used in student t distribution - not used
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:returns: gradient of likelihood evaluated at points
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:rtype: Nx1 array
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"""
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assert y.shape == f.shape
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e = y - f
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assert y.shape == link_f.shape
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e = y - link_f
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grad = ((self.v + 1) * e) / (self.v * self.sigma2 + (e**2))
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return grad
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def d2lik_d2f(self, y, f, extra_data=None):
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def d2logpdf_dlink2(self, link_f, y, extra_data=None):
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"""
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Hessian at y, given f, w.r.t f the hessian will be 0 unless i == j
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Hessian at y, given link(f), w.r.t link(f) the hessian will be 0 unless i == j
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i.e. second derivative lik_function at y given f_{i} f_{j} w.r.t f_{i} and f_{j}
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.. math::
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\\frac{d^{2} \\ln p(y_{i}|f_{i})}{d^{2}f} = \\frac{(v+1)((y_{i}-f_{i})^{2} - \\sigma^{2}v)}{((y_{i}-f_{i})^{2} + \\sigma^{2}v)^{2}}
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:param link_f: latent variables link(f)
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:type link_f: Nx1 array
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:param y: data
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:type y: Nx1 array
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:param f: latent variables f
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:type f: Nx1 array
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:param extra_data: extra_data which is not used in student t distribution - not used
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:returns: Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)
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:rtype: Nx1 array
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@ -106,101 +124,101 @@ class StudentT(NoiseDistribution):
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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 f_{i} not on f_{j!=i}
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"""
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assert y.shape == f.shape
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e = y - f
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assert y.shape == link_f.shape
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e = y - link_f
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hess = ((self.v + 1)*(e**2 - self.v*self.sigma2)) / ((self.sigma2*self.v + e**2)**2)
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return hess
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def d3lik_d3f(self, y, f, extra_data=None):
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def d3logpdf_dlink3(self, link_f, y, extra_data=None):
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"""
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Third order derivative log-likelihood function at y given f w.r.t f
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.. math::
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\\frac{d^{3} \\ln p(y_{i}|f_{i})}{d^{3}f} = \\frac{-2(v+1)((y_{i} - f_{i})^3 - 3(y_{i} - f_{i}) \\sigma^{2} v))}{((y_{i} - f_{i}) + \\sigma^{2} v)^3}
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:param link_f: latent variables link(f)
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:type link_f: Nx1 array
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:param y: data
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:type y: Nx1 array
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:param f: latent variables f
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:type f: Nx1 array
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:param extra_data: extra_data which is not used in student t distribution - not used
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:returns: third derivative of likelihood evaluated at points f
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:rtype: Nx1 array
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"""
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assert y.shape == f.shape
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e = y - f
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d3lik_d3f = ( -(2*(self.v + 1)*(-e)*(e**2 - 3*self.v*self.sigma2)) /
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assert y.shape == link_f.shape
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e = y - link_f
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d3lik_dlink3 = ( -(2*(self.v + 1)*(-e)*(e**2 - 3*self.v*self.sigma2)) /
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((e**2 + self.sigma2*self.v)**3)
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)
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return d3lik_d3f
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return d3lik_dlink3
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def dlik_dvar(self, y, f, extra_data=None):
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def dlogpdf_dvar(self, link_f, y, extra_data=None):
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"""
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Gradient of the log-likelihood function at y given f, w.r.t variance parameter (t_noise)
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.. math::
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\\frac{d \\ln p(y_{i}|f_{i})}{d\\sigma^{2}} = \\frac{v((y_{i} - f_{i})^{2} - \\sigma^{2})}{2\\sigma^{2}(\\sigma^{2}v + (y_{i} - f_{i})^{2})}
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:param link_f: latent variables link(f)
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:type link_f: Nx1 array
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:param y: data
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:type y: Nx1 array
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:param f: latent variables f
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:type f: Nx1 array
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:param extra_data: extra_data which is not used in student t distribution - not used
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:returns: derivative of likelihood evaluated at points f w.r.t variance parameter
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:rtype: float
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"""
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assert y.shape == f.shape
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e = y - f
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dlik_dvar = self.v*(e**2 - self.sigma2)/(2*self.sigma2*(self.sigma2*self.v + e**2))
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#FIXME: May not want to sum over all dimensions if using many D?
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return np.sum(dlik_dvar)
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assert y.shape == link_f.shape
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e = y - link_f
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dlogpdf_dvar = self.v*(e**2 - self.sigma2)/(2*self.sigma2*(self.sigma2*self.v + e**2))
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#FIXME: Careful as this hasn't been chained with dlink_var, not sure if we want link functions on our parameters?! Shouldn't need them with constraints
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return np.sum(dlogpdf_dvar)
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def dlik_df_dvar(self, y, f, extra_data=None):
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def dlogpdf_dlink_dvar(self, link_f, y, extra_data=None):
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"""
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Derivative of the dlik_df w.r.t variance parameter (t_noise)
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Derivative of the dlogpdf_dlink w.r.t variance parameter (t_noise)
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.. math::
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\\frac{d}{d\\sigma^{2}}(\\frac{d \\ln p(y_{i}|f_{i})}{df}) = \\frac{-2\\sigma v(v + 1)(y_{i}-f_{i})}{(y_{i}-f_{i})^2 + \\sigma^2 v)^2}
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:param link_f: latent variables link_f
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:type link_f: Nx1 array
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:param y: data
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:type y: Nx1 array
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:param f: latent variables f
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:type f: Nx1 array
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:param extra_data: extra_data which is not used in student t distribution - not used
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:returns: derivative of likelihood evaluated at points f w.r.t variance parameter
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:rtype: Nx1 array
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"""
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assert y.shape == f.shape
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e = y - f
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dlik_grad_dvar = (self.v*(self.v+1)*(-e))/((self.sigma2*self.v + e**2)**2)
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return dlik_grad_dvar
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assert y.shape == link_f.shape
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e = y - link_f
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dlogpdf_dlink_dvar = (self.v*(self.v+1)*(-e))/((self.sigma2*self.v + e**2)**2)
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return dlogpdf_dlink_dvar
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def d2lik_d2f_dvar(self, y, f, extra_data=None):
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def d2logpdf_dlink2_dvar(self, link_f, y, extra_data=None):
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"""
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Gradient of the hessian (d2lik_d2f) w.r.t variance parameter (t_noise)
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Gradient of the hessian (d2logpdf_dlink2) w.r.t variance parameter (t_noise)
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.. math::
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\\frac{d}{d\\sigma^{2}}(\\frac{d^{2} \\ln p(y_{i}|f_{i})}{d^{2}f}) = \\frac{v(v+1)(\\sigma^{2}v - 3(y_{i} - f_{i})^{2})}{(\\sigma^{2}v + (y_{i} - f_{i})^{2})^{3}}
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:param link_f: latent variables link(f)
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:type link_f: Nx1 array
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:param y: data
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:type y: Nx1 array
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:param f: latent variables f
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:type f: Nx1 array
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:param extra_data: extra_data which is not used in student t distribution - not used
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:returns: derivative of hessian evaluated at points f and f_j w.r.t variance parameter
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:rtype: Nx1 array
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"""
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assert y.shape == f.shape
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e = y - f
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dlik_hess_dvar = ( (self.v*(self.v+1)*(self.sigma2*self.v - 3*(e**2)))
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assert y.shape == link_f.shape
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e = y - link_f
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d2logpdf_dlink2_dvar = ( (self.v*(self.v+1)*(self.sigma2*self.v - 3*(e**2)))
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/ ((self.sigma2*self.v + (e**2))**3)
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)
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return dlik_hess_dvar
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return d2logpdf_dlink2_dvar
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def _laplace_gradients(self, y, f, extra_data=None):
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#must be listed in same order as 'get_param_names'
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derivs = ([self.dlik_dvar(y, f, extra_data=extra_data)],
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[self.dlik_df_dvar(y, f, extra_data=extra_data)],
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[self.d2lik_d2f_dvar(y, f, extra_data=extra_data)]
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derivs = ([self.dlogpdf_dvar(f, y, extra_data=extra_data)],
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[self.dlogpdf_dlink_dvar(f, y, extra_data=extra_data)],
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[self.d2logpdf_dlink2_dvar(f, y, extra_data=extra_data)]
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) # lists as we might learn many parameters
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# ensure we have gradients for every parameter we want to optimize
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assert len(derivs[0]) == len(self._get_param_names())
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