GPy.likelihoods package¶

Submodules¶

GPy.likelihoods.bernoulli module¶

class GPy.likelihoods.bernoulli.Bernoulli(gp_link=None)[source]¶

Bases: GPy.likelihoods.likelihood.Likelihood

Bernoulli likelihood

$p(y_{i}|\lambda(f_{i})) = \lambda(f_{i})^{y_{i}}(1-f_{i})^{1-y_{i}}$

Note

Y takes values in either {-1, 1} or {0, 1}. link function should have the domain [0, 1], e.g. probit (default) or Heaviside

d2logpdf_dlink2(inv_link_f, y, Y_metadata=None)[source]¶

Hessian at y, given inv_link_f, w.r.t inv_link_f the hessian will be 0 unless i == j 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.

$\frac{d^{2}\ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)^{2}} = \frac{-y_{i}}{\lambda(f)^{2}} - \frac{(1-y_{i})}{(1-\lambda(f))^{2}}$

Parameters:	inv_link_f (Nx1 array) – latent variables inverse link of f. y (Nx1 array) – data Y_metadata – Y_metadata not used in bernoulli
Returns:	Diagonal of log hessian matrix (second derivative of log likelihood evaluated at points inverse link of f.
Return type:	Nx1 array

Note

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 inverse link of f_i not on inverse link of f_(j!=i)

d3logpdf_dlink3(inv_link_f, y, Y_metadata=None)[source]¶

Third order derivative log-likelihood function at y given inverse link of f w.r.t inverse link of f

$\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = \frac{2y_{i}}{\lambda(f)^{3}} - \frac{2(1-y_{i}}{(1-\lambda(f))^{3}}$

Parameters:	inv_link_f (Nx1 array) – latent variables passed through inverse link of f. y (Nx1 array) – data Y_metadata – Y_metadata not used in bernoulli
Returns:	third derivative of log likelihood evaluated at points inverse_link(f)
Return type:	Nx1 array

dlogpdf_dlink(inv_link_f, y, Y_metadata=None)[source]¶

Gradient of the pdf at y, given inverse link of f w.r.t inverse link of f.

$\frac{d\ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \frac{y_{i}}{\lambda(f_{i})} - \frac{(1 - y_{i})}{(1 - \lambda(f_{i}))}$

Parameters:	inv_link_f (Nx1 array) – latent variables inverse link of f. y (Nx1 array) – data Y_metadata – Y_metadata not used in bernoulli
Returns:	gradient of log likelihood evaluated at points inverse link of f.
Return type:	Nx1 array

exact_inference_gradients(dL_dKdiag, Y_metadata=None)[source]¶

logpdf_link(inv_link_f, y, Y_metadata=None)[source]¶

Log Likelihood function given inverse link of f.

$\ln p(y_{i}|\lambda(f_{i})) = y_{i}\log\lambda(f_{i}) + (1-y_{i})\log (1-f_{i})$

Parameters:	inv_link_f (Nx1 array) – latent variables inverse link of f. y (Nx1 array) – data Y_metadata – Y_metadata not used in bernoulli
Returns:	log likelihood evaluated at points inverse link of f.
Return type:	float

moments_match_ep(Y_i, tau_i, v_i)[source]¶

Moments match of the marginal approximation in EP algorithm

Parameters:	i – number of observation (int) tau_i – precision of the cavity distribution (float) v_i – mean/variance of the cavity distribution (float)

pdf_link(inv_link_f, y, Y_metadata=None)[source]¶

Likelihood function given inverse link of f.

$p(y_{i}|\lambda(f_{i})) = \lambda(f_{i})^{y_{i}}(1-f_{i})^{1-y_{i}}$

Parameters:	inv_link_f (Nx1 array) – latent variables inverse link of f. y (Nx1 array) – data Y_metadata – Y_metadata not used in bernoulli
Returns:	likelihood evaluated for this point
Return type:	float

predictive_mean(mu, variance, Y_metadata=None)[source]¶

predictive_variance(mu, variance, pred_mean, Y_metadata=None)[source]¶

samples(gp, Y_metadata=None)[source]¶

Returns a set of samples of observations based on a given value of the latent variable.

Parameters:	gp – latent variable

GPy.likelihoods.exponential module¶

class GPy.likelihoods.exponential.Exponential(gp_link=None)[source]¶

Bases: GPy.likelihoods.likelihood.Likelihood

Expoential likelihood Y is expected to take values in {0,1,2,...} —– $$ L(x) = exp(lambda) * lambda**Y_i / Y_i! $$

d2logpdf_dlink2(link_f, y, Y_metadata=None)[source]¶

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

$\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}\lambda(f)} = -\frac{1}{\lambda(f_{i})^{2}}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in exponential distribution
Returns:	Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)
Return type:	Nx1 array

Note

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))

d3logpdf_dlink3(link_f, y, Y_metadata=None)[source]¶

Third order derivative log-likelihood function at y given link(f) w.r.t link(f)

$\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = \frac{2}{\lambda(f_{i})^{3}}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in exponential distribution
Returns:	third derivative of likelihood evaluated at points f
Return type:	Nx1 array

dlogpdf_dlink(link_f, y, Y_metadata=None)[source]¶

Gradient of the log likelihood function at y, given link(f) w.r.t link(f)

$\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \frac{1}{\lambda(f)} - y_{i}$

Parameters:	link_f (Nx1 array) – latent variables (f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in exponential distribution
Returns:	gradient of likelihood evaluated at points
Return type:	Nx1 array

logpdf_link(link_f, y, Y_metadata=None)[source]¶

Log Likelihood Function given link(f)

$\ln p(y_{i}|\lambda(f_{i})) = \ln \lambda(f_{i}) - y_{i}\lambda(f_{i})$

Parameters:	link_f (Nx1 array) – latent variables (link(f)) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in exponential distribution
Returns:	likelihood evaluated for this point
Return type:	float

pdf_link(link_f, y, Y_metadata=None)[source]¶

Likelihood function given link(f)

$p(y_{i}|\lambda(f_{i})) = \lambda(f_{i})\exp (-y\lambda(f_{i}))$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in exponential distribution
Returns:	likelihood evaluated for this point
Return type:	float

samples(gp)[source]¶

Returns a set of samples of observations based on a given value of the latent variable.

Parameters:	gp – latent variable

GPy.likelihoods.gamma module¶

class GPy.likelihoods.gamma.Gamma(gp_link=None, beta=1.0)[source]¶

Bases: GPy.likelihoods.likelihood.Likelihood

Gamma likelihood

$p(y_{i}|\lambda(f_{i})) = \frac{\beta^{\alpha_{i}}}{\Gamma(\alpha_{i})}y_{i}^{\alpha_{i}-1}e^{-\beta y_{i}}\\ \alpha_{i} = \beta y_{i}$

d2logpdf_dlink2(link_f, y, Y_metadata=None)[source]¶

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

$\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}\lambda(f)} = -\beta^{2}\frac{d\Psi(\alpha_{i})}{d\alpha_{i}}\\ \alpha_{i} = \beta y_{i}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in gamma distribution
Returns:	Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)
Return type:	Nx1 array

Note

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))

d3logpdf_dlink3(link_f, y, Y_metadata=None)[source]¶

Third order derivative log-likelihood function at y given link(f) w.r.t link(f)

$\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = -\beta^{3}\frac{d^{2}\Psi(\alpha_{i})}{d\alpha_{i}}\\ \alpha_{i} = \beta y_{i}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in gamma distribution
Returns:	third derivative of likelihood evaluated at points f
Return type:	Nx1 array

dlogpdf_dlink(link_f, y, Y_metadata=None)[source]¶

Gradient of the log likelihood function at y, given link(f) w.r.t link(f)

$\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \beta (\log \beta y_{i}) - \Psi(\alpha_{i})\beta\\ \alpha_{i} = \beta y_{i}$

Parameters:	link_f (Nx1 array) – latent variables (f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in gamma distribution
Returns:	gradient of likelihood evaluated at points
Return type:	Nx1 array

logpdf_link(link_f, y, Y_metadata=None)[source]¶

Log Likelihood Function given link(f)

$\ln p(y_{i}|\lambda(f_{i})) = \alpha_{i}\log \beta - \log \Gamma(\alpha_{i}) + (\alpha_{i} - 1)\log y_{i} - \beta y_{i}\\ \alpha_{i} = \beta y_{i}$

Parameters:	link_f (Nx1 array) – latent variables (link(f)) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in poisson distribution
Returns:	likelihood evaluated for this point
Return type:	float

pdf_link(link_f, y, Y_metadata=None)[source]¶

Likelihood function given link(f)

$p(y_{i}|\lambda(f_{i})) = \frac{\beta^{\alpha_{i}}}{\Gamma(\alpha_{i})}y_{i}^{\alpha_{i}-1}e^{-\beta y_{i}}\\ \alpha_{i} = \beta y_{i}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in poisson distribution
Returns:	likelihood evaluated for this point
Return type:	float

GPy.likelihoods.gaussian module¶

A lot of this code assumes that the link function is the identity.

I think laplace code is okay, but I’m quite sure that the EP moments will only work if the link is identity.

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.

James 11/12/13

class GPy.likelihoods.gaussian.Gaussian(gp_link=None, variance=1.0, name='Gaussian_noise')[source]¶

Bases: GPy.likelihoods.likelihood.Likelihood

Gaussian likelihood

$\ln p(y_{i}|\lambda(f_{i})) = -\frac{N \ln 2\pi}{2} - \frac{\ln |K|}{2} - \frac{(y_{i} - \lambda(f_{i}))^{T}\sigma^{-2}(y_{i} - \lambda(f_{i}))}{2}$

Parameters:	variance – variance value of the Gaussian distribution N (int) – Number of data points

betaY(Y, Y_metadata=None)[source]¶

d2logpdf_dlink2(link_f, y, Y_metadata=None)[source]¶

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)

The hessian will be 0 unless i == j

$\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}f} = -\frac{1}{\sigma^{2}}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata not used in gaussian
Returns:	Diagonal of log hessian matrix (second derivative of log likelihood evaluated at points link(f))
Return type:	Nx1 array

Note

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))

d2logpdf_dlink2_dtheta(f, y, Y_metadata=None)[source]¶

d2logpdf_dlink2_dvar(link_f, y, Y_metadata=None)[source]¶

Gradient of the hessian (d2logpdf_dlink2) w.r.t variance parameter (noise_variance)

$\frac{d}{d\sigma^{2}}(\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}\lambda(f)}) = \frac{1}{\sigma^{4}}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata not used in gaussian
Returns:	derivative of log hessian evaluated at points link(f_i) and link(f_j) w.r.t variance parameter
Return type:	Nx1 array

d3logpdf_dlink3(link_f, y, Y_metadata=None)[source]¶

Third order derivative log-likelihood function at y given link(f) w.r.t link(f)

$\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = 0$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata not used in gaussian
Returns:	third derivative of log likelihood evaluated at points link(f)
Return type:	Nx1 array

dlogpdf_dlink(link_f, y, Y_metadata=None)[source]¶

Gradient of the pdf at y, given link(f) w.r.t link(f)

$\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \frac{1}{\sigma^{2}}(y_{i} - \lambda(f_{i}))$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata not used in gaussian
Returns:	gradient of log likelihood evaluated at points link(f)
Return type:	Nx1 array

dlogpdf_dlink_dtheta(f, y, Y_metadata=None)[source]¶

dlogpdf_dlink_dvar(link_f, y, Y_metadata=None)[source]¶

Derivative of the dlogpdf_dlink w.r.t variance parameter (noise_variance)

$\frac{d}{d\sigma^{2}}(\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)}) = \frac{1}{\sigma^{4}}(-y_{i} + \lambda(f_{i}))$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata not used in gaussian
Returns:	derivative of log likelihood evaluated at points link(f) w.r.t variance parameter
Return type:	Nx1 array

dlogpdf_link_dtheta(f, y, Y_metadata=None)[source]¶

dlogpdf_link_dvar(link_f, y, Y_metadata=None)[source]¶

Gradient of the log-likelihood function at y given link(f), w.r.t variance parameter (noise_variance)

$\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\sigma^{2}} = -\frac{N}{2\sigma^{2}} + \frac{(y_{i} - \lambda(f_{i}))^{2}}{2\sigma^{4}}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata not used in gaussian
Returns:	derivative of log likelihood evaluated at points link(f) w.r.t variance parameter
Return type:	float

exact_inference_gradients(dL_dKdiag, Y_metadata=None)[source]¶

gaussian_variance(Y_metadata=None)[source]¶

log_predictive_density(y_test, mu_star, var_star)[source]¶: assumes independence

logpdf_link(link_f, y, Y_metadata=None)[source]¶

Log likelihood function given link(f)

$\ln p(y_{i}|\lambda(f_{i})) = -\frac{N \ln 2\pi}{2} - \frac{\ln |K|}{2} - \frac{(y_{i} - \lambda(f_{i}))^{T}\sigma^{-2}(y_{i} - \lambda(f_{i}))}{2}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata not used in gaussian
Returns:	log likelihood evaluated for this point
Return type:	float

pdf_link(link_f, y, Y_metadata=None)[source]¶

Likelihood function given link(f)

$\ln p(y_{i}|\lambda(f_{i})) = -\frac{N \ln 2\pi}{2} - \frac{\ln |K|}{2} - \frac{(y_{i} - \lambda(f_{i}))^{T}\sigma^{-2}(y_{i} - \lambda(f_{i}))}{2}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata not used in gaussian
Returns:	likelihood evaluated for this point
Return type:	float

predictive_mean(mu, sigma)[source]¶

predictive_quantiles(mu, var, quantiles, Y_metadata=None)[source]¶

predictive_values(mu, var, full_cov=False, Y_metadata=None)[source]¶

predictive_variance(mu, sigma, predictive_mean=None)[source]¶

samples(gp, Y_metadata=None)[source]¶

Returns a set of samples of observations based on a given value of the latent variable.

Parameters:	gp – latent variable

update_gradients(grad)[source]¶

GPy.likelihoods.likelihood module¶

class GPy.likelihoods.likelihood.Likelihood(gp_link, name)[source]¶

Bases: GPy.core.parameterization.parameterized.Parameterized

Likelihood base class, used to defing p(y|f).

All instances use _inverse_ link functions, which can be swapped out. It is expected that inheriting classes define a default inverse link function

To use this class, inherit and define missing functionality.

Inheriting classes must implement:: pdf_link : a bound method which turns the output of the link function into the pdf logpdf_link : the logarithm of the above
To enable use with EP, inheriting classes must define:: TODO: a suitable derivative function for any parameters of the class
It is also desirable to define:: moments_match_ep : a function to compute the EP moments If this isn’t defined, the moments will be computed using 1D quadrature.
To enable use with Laplace approximation, inheriting classes must define:: Some derivative functions AS TODO

For exact Gaussian inference, define JH TODO

conditional_mean(gp)[source]¶: The mean of the random variable conditioned on one value of the GP

conditional_variance(gp)[source]¶: The variance of the random variable conditioned on one value of the GP

d2logpdf_df2(f, y, Y_metadata=None)[source]¶

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

$\frac{d^{2}\log p(y|\lambda(f))}{df^{2}} = \frac{d^{2}\log p(y|\lambda(f))}{d^{2}\lambda(f)}\left(\frac{d\lambda(f)}{df}\right)^{2} + \frac{d\log p(y|\lambda(f))}{d\lambda(f)}\frac{d^{2}\lambda(f)}{df^{2}}$

Parameters:	f (Nx1 array) – latent variables f y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution - not used
Returns:	second derivative of log likelihood evaluated for this point (diagonal only)
Return type:	1xN array

d2logpdf_df2_dtheta(f, y, Y_metadata=None)[source]¶: TODO: Doc strings

d2logpdf_dlink2(inv_link_f, y, Y_metadata=None)[source]¶

d2logpdf_dlink2_dtheta(inv_link_f, y, Y_metadata=None)[source]¶

d3logpdf_df3(f, y, Y_metadata=None)[source]¶

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

$\frac{d^{3}\log p(y|\lambda(f))}{df^{3}} = \frac{d^{3}\log p(y|\lambda(f)}{d\lambda(f)^{3}}\left(\frac{d\lambda(f)}{df}\right)^{3} + 3\frac{d^{2}\log p(y|\lambda(f)}{d\lambda(f)^{2}}\frac{d\lambda(f)}{df}\frac{d^{2}\lambda(f)}{df^{2}} + \frac{d\log p(y|\lambda(f)}{d\lambda(f)}\frac{d^{3}\lambda(f)}{df^{3}}$

Parameters:	f (Nx1 array) – latent variables f y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution - not used
Returns:	third derivative of log likelihood evaluated for this point
Return type:	float

d3logpdf_dlink3(inv_link_f, y, Y_metadata=None)[source]¶

dlogpdf_df(f, y, Y_metadata=None)[source]¶

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

$\frac{d\log p(y|\lambda(f))}{df} = \frac{d\log p(y|\lambda(f))}{d\lambda(f)}\frac{d\lambda(f)}{df}$

Parameters:	f (Nx1 array) – latent variables f y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution - not used
Returns:	derivative of log likelihood evaluated for this point
Return type:	1xN array

dlogpdf_df_dtheta(f, y, Y_metadata=None)[source]¶: TODO: Doc strings

dlogpdf_dlink(inv_link_f, y, Y_metadata=None)[source]¶

dlogpdf_dlink_dtheta(inv_link_f, y, Y_metadata=None)[source]¶

dlogpdf_dtheta(f, y, Y_metadata=None)[source]¶: TODO: Doc strings

dlogpdf_link_dtheta(inv_link_f, y, Y_metadata=None)[source]¶

log_predictive_density(y_test, mu_star, var_star)[source]¶

Calculation of the log predictive density

Parameters:	y_test ((Nx1) array) – test observations (y_{}) mu_star* ((Nx1) array) – predictive mean of gaussian p(f_{}\|mu_{}, var_{}) var_star* ((Nx1) array) – predictive variance of gaussian p(f_{}\|mu_{}, var_{*})

logpdf(f, y, Y_metadata=None)[source]¶

Evaluates the link function link(f) then computes the log likelihood (log pdf) using it

Parameters:	f (Nx1 array) – latent variables f y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution - not used
Returns:	log likelihood evaluated for this point
Return type:	float

logpdf_link(inv_link_f, y, Y_metadata=None)[source]¶

pdf(f, y, Y_metadata=None)[source]¶

Evaluates the link function link(f) then computes the likelihood (pdf) using it

Parameters:	f (Nx1 array) – latent variables f y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution - not used
Returns:	likelihood evaluated for this point
Return type:	float

pdf_link(inv_link_f, y, Y_metadata=None)[source]¶

predictive_mean(mu, variance, Y_metadata=None)[source]¶

Quadrature calculation of the predictive mean: E(Y_star|Y) = E( E(Y_star|f_star, Y) )

Parameters:	mu – mean of posterior sigma – standard deviation of posterior

predictive_quantiles(mu, var, quantiles, Y_metadata=None)[source]¶

predictive_values(mu, var, full_cov=False, Y_metadata=None)[source]¶

Compute mean, variance of the predictive distibution.

Parameters:	mu – mean of the latent variable, f, of posterior var – variance of the latent variable, f, of posterior full_cov (Boolean) – whether to use the full covariance or just the diagonal

predictive_variance(mu, variance, predictive_mean=None, Y_metadata=None)[source]¶

Approximation to the predictive variance: V(Y_star)

The following variance decomposition is used: V(Y_star) = E( V(Y_star|f_star) ) + V( E(Y_star|f_star) )

Predictive_mean:
Parameters:	mu – mean of posterior sigma – standard deviation of posterior
	output’s predictive mean, if None _predictive_mean function will be called.

samples(gp, Y_metadata=None)[source]¶

Returns a set of samples of observations based on a given value of the latent variable.

Parameters:	gp – latent variable

update_gradients(partial)[source]¶

variational_expectations(Y, m, v, gh_points=None)[source]¶

Use Gauss-Hermite Quadrature to compute

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) ]

where p(f) is a Gaussian with mean m and variance v. The shapes of Y, m and v should match.

if no gh_points are passed, we construct them using defualt options

GPy.likelihoods.link_functions module¶

class GPy.likelihoods.link_functions.Cloglog[source]¶

Bases: GPy.likelihoods.link_functions.GPTransformation

Complementary log-log link .. math:

p(f) = 1 - e^{-e^f}

or

f = \log (-\log(1-p))

d2transf_df2(f)[source]¶

d3transf_df3(f)[source]¶

dtransf_df(f)[source]¶

transf(f)[source]¶

class GPy.likelihoods.link_functions.GPTransformation[source]¶

Bases: object

Link function class for doing non-Gaussian likelihoods approximation

Parameters:	Y – observed output (Nx1 numpy.darray)

Note

Y values allowed depend on the likelihood_function used

d2transf_df2(f)[source]¶: second derivative of transf(f) w.r.t. f

d3transf_df3(f)[source]¶: third derivative of transf(f) w.r.t. f

dtransf_df(f)[source]¶: derivative of transf(f) w.r.t. f

transf(f)[source]¶: Gaussian process tranformation function, latent space -> output space

class GPy.likelihoods.link_functions.Heaviside[source]¶

Bases: GPy.likelihoods.link_functions.GPTransformation

$g(f) = I_{x \in A}$

d2transf_df2(f)[source]¶

dtransf_df(f)[source]¶

transf(f)[source]¶

class GPy.likelihoods.link_functions.Identity[source]¶

Bases: GPy.likelihoods.link_functions.GPTransformation

$g(f) = f$

d2transf_df2(f)[source]¶

d3transf_df3(f)[source]¶

dtransf_df(f)[source]¶

transf(f)[source]¶

class GPy.likelihoods.link_functions.Log[source]¶

Bases: GPy.likelihoods.link_functions.GPTransformation

$g(f) = \log(\mu)$

d2transf_df2(f)[source]¶

d3transf_df3(f)[source]¶

dtransf_df(f)[source]¶

transf(f)[source]¶

class GPy.likelihoods.link_functions.Log_ex_1[source]¶

Bases: GPy.likelihoods.link_functions.GPTransformation

$g(f) = \log(\exp(\mu) - 1)$

d2transf_df2(f)[source]¶

d3transf_df3(f)[source]¶

dtransf_df(f)[source]¶

transf(f)[source]¶

class GPy.likelihoods.link_functions.Probit[source]¶

Bases: GPy.likelihoods.link_functions.GPTransformation

$g(f) = \Phi^{-1} (mu)$

d2transf_df2(f)[source]¶

d3transf_df3(f)[source]¶

dtransf_df(f)[source]¶

transf(f)[source]¶

class GPy.likelihoods.link_functions.Reciprocal[source]¶

Bases: GPy.likelihoods.link_functions.GPTransformation

d2transf_df2(f)[source]¶

d3transf_df3(f)[source]¶

dtransf_df(f)[source]¶

transf(f)[source]¶

GPy.likelihoods.mixed_noise module¶

class GPy.likelihoods.mixed_noise.MixedNoise(likelihoods_list, name='mixed_noise')[source]¶

Bases: GPy.likelihoods.likelihood.Likelihood

betaY(Y, Y_metadata)[source]¶

exact_inference_gradients(dL_dKdiag, Y_metadata)[source]¶

gaussian_variance(Y_metadata)[source]¶

predictive_quantiles(mu, var, quantiles, Y_metadata)[source]¶

predictive_values(mu, var, full_cov=False, Y_metadata=None)[source]¶

predictive_variance(mu, sigma, Y_metadata)[source]¶

samples(gp, Y_metadata)[source]¶

Returns a set of samples of observations based on a given value of the latent variable.

Parameters:	gp – latent variable

update_gradients(gradients)[source]¶

GPy.likelihoods.poisson module¶

class GPy.likelihoods.poisson.Poisson(gp_link=None)[source]¶

Bases: GPy.likelihoods.likelihood.Likelihood

Poisson likelihood

$p(y_{i}|\lambda(f_{i})) = \frac{\lambda(f_{i})^{y_{i}}}{y_{i}!}e^{-\lambda(f_{i})}$

Note

Y is expected to take values in {0,1,2,...}

conditional_mean(gp)[source]¶: The mean of the random variable conditioned on one value of the GP

conditional_variance(gp)[source]¶: The variance of the random variable conditioned on one value of the GP

d2logpdf_dlink2(link_f, y, Y_metadata=None)[source]¶

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

$\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}\lambda(f)} = \frac{-y_{i}}{\lambda(f_{i})^{2}}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in poisson distribution
Returns:	Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)
Return type:	Nx1 array

Note

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))

d3logpdf_dlink3(link_f, y, Y_metadata=None)[source]¶

Third order derivative log-likelihood function at y given link(f) w.r.t link(f)

$\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = \frac{2y_{i}}{\lambda(f_{i})^{3}}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in poisson distribution
Returns:	third derivative of likelihood evaluated at points f
Return type:	Nx1 array

dlogpdf_dlink(link_f, y, Y_metadata=None)[source]¶

Gradient of the log likelihood function at y, given link(f) w.r.t link(f)

$\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \frac{y_{i}}{\lambda(f_{i})} - 1$

Parameters:	link_f (Nx1 array) – latent variables (f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in poisson distribution
Returns:	gradient of likelihood evaluated at points
Return type:	Nx1 array

logpdf_link(link_f, y, Y_metadata=None)[source]¶

Log Likelihood Function given link(f)

$\ln p(y_{i}|\lambda(f_{i})) = -\lambda(f_{i}) + y_{i}\log \lambda(f_{i}) - \log y_{i}!$

Parameters:	link_f (Nx1 array) – latent variables (link(f)) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in poisson distribution
Returns:	likelihood evaluated for this point
Return type:	float

pdf_link(link_f, y, Y_metadata=None)[source]¶

Likelihood function given link(f)

$p(y_{i}|\lambda(f_{i})) = \frac{\lambda(f_{i})^{y_{i}}}{y_{i}!}e^{-\lambda(f_{i})}$

Parameters:	link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in poisson distribution
Returns:	likelihood evaluated for this point
Return type:	float

samples(gp, Y_metadata=None)[source]¶

Returns a set of samples of observations based on a given value of the latent variable.

Parameters:	gp – latent variable

GPy.likelihoods.student_t module¶

class GPy.likelihoods.student_t.StudentT(gp_link=None, deg_free=5, sigma2=2)[source]¶

Bases: GPy.likelihoods.likelihood.Likelihood

Student T likelihood

For nomanclature see Bayesian Data Analysis 2003 p576

$p(y_{i}|\lambda(f_{i})) = \frac{\Gamma\left(\frac{v+1}{2}\right)}{\Gamma\left(\frac{v}{2}\right)\sqrt{v\pi\sigma^{2}}}\left(1 + \frac{1}{v}\left(\frac{(y_{i} - f_{i})^{2}}{\sigma^{2}}\right)\right)^{\frac{-v+1}{2}}$

conditional_mean(gp)[source]¶

conditional_variance(gp)[source]¶

d2logpdf_dlink2(inv_link_f, y, Y_metadata=None)[source]¶

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

$\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}\lambda(f)} = \frac{(v+1)((y_{i}-\lambda(f_{i}))^{2} - \sigma^{2}v)}{((y_{i}-\lambda(f_{i}))^{2} + \sigma^{2}v)^{2}}$

Parameters:	inv_link_f (Nx1 array) – latent variables inv_link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution
Returns:	Diagonal of hessian matrix (second derivative of likelihood evaluated at points f)
Return type:	Nx1 array

Note

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))

d2logpdf_dlink2_dtheta(f, y, Y_metadata=None)[source]¶

d2logpdf_dlink2_dvar(inv_link_f, y, Y_metadata=None)[source]¶

Gradient of the hessian (d2logpdf_dlink2) w.r.t variance parameter (t_noise)

$\frac{d}{d\sigma^{2}}(\frac{d^{2} \ln p(y_{i}|\lambda(f_{i}))}{d^{2}f}) = \frac{v(v+1)(\sigma^{2}v - 3(y_{i} - \lambda(f_{i}))^{2})}{(\sigma^{2}v + (y_{i} - \lambda(f_{i}))^{2})^{3}}$

Parameters:	inv_link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution
Returns:	derivative of hessian evaluated at points f and f_j w.r.t variance parameter
Return type:	Nx1 array

d3logpdf_dlink3(inv_link_f, y, Y_metadata=None)[source]¶

Third order derivative log-likelihood function at y given link(f) w.r.t link(f)

$\frac{d^{3} \ln p(y_{i}|\lambda(f_{i}))}{d^{3}\lambda(f)} = \frac{-2(v+1)((y_{i} - \lambda(f_{i}))^3 - 3(y_{i} - \lambda(f_{i})) \sigma^{2} v))}{((y_{i} - \lambda(f_{i})) + \sigma^{2} v)^3}$

Parameters:	inv_link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution
Returns:	third derivative of likelihood evaluated at points f
Return type:	Nx1 array

dlogpdf_dlink(inv_link_f, y, Y_metadata=None)[source]¶

Gradient of the log likelihood function at y, given link(f) w.r.t link(f)

$\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\lambda(f)} = \frac{(v+1)(y_{i}-\lambda(f_{i}))}{(y_{i}-\lambda(f_{i}))^{2} + \sigma^{2}v}$

Parameters:	inv_link_f (Nx1 array) – latent variables (f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution
Returns:	gradient of likelihood evaluated at points
Return type:	Nx1 array

dlogpdf_dlink_dtheta(f, y, Y_metadata=None)[source]¶

dlogpdf_dlink_dvar(inv_link_f, y, Y_metadata=None)[source]¶

Derivative of the dlogpdf_dlink w.r.t variance parameter (t_noise)

$\frac{d}{d\sigma^{2}}(\frac{d \ln p(y_{i}|\lambda(f_{i}))}{df}) = \frac{-2\sigma v(v + 1)(y_{i}-\lambda(f_{i}))}{(y_{i}-\lambda(f_{i}))^2 + \sigma^2 v)^2}$

Parameters:	inv_link_f (Nx1 array) – latent variables inv_link_f y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution
Returns:	derivative of likelihood evaluated at points f w.r.t variance parameter
Return type:	Nx1 array

dlogpdf_link_dtheta(f, y, Y_metadata=None)[source]¶

dlogpdf_link_dvar(inv_link_f, y, Y_metadata=None)[source]¶

Gradient of the log-likelihood function at y given f, w.r.t variance parameter (t_noise)

$\frac{d \ln p(y_{i}|\lambda(f_{i}))}{d\sigma^{2}} = \frac{v((y_{i} - \lambda(f_{i}))^{2} - \sigma^{2})}{2\sigma^{2}(\sigma^{2}v + (y_{i} - \lambda(f_{i}))^{2})}$

Parameters:	inv_link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution
Returns:	derivative of likelihood evaluated at points f w.r.t variance parameter
Return type:	float

logpdf_link(inv_link_f, y, Y_metadata=None)[source]¶

Log Likelihood Function given link(f)

$\ln p(y_{i}|\lambda(f_{i})) = \ln \Gamma\left(\frac{v+1}{2}\right) - \ln \Gamma\left(\frac{v}{2}\right) - \ln \sqrt{v \pi\sigma^{2}} - \frac{v+1}{2}\ln \left(1 + \frac{1}{v}\left(\frac{(y_{i} - \lambda(f_{i}))^{2}}{\sigma^{2}}\right)\right)$

Parameters:	inv_link_f (Nx1 array) – latent variables (link(f)) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution
Returns:	likelihood evaluated for this point
Return type:	float

parameters_changed()[source]¶

pdf_link(inv_link_f, y, Y_metadata=None)[source]¶

Likelihood function given link(f)

$p(y_{i}|\lambda(f_{i})) = \frac{\Gamma\left(\frac{v+1}{2}\right)}{\Gamma\left(\frac{v}{2}\right)\sqrt{v\pi\sigma^{2}}}\left(1 + \frac{1}{v}\left(\frac{(y_{i} - \lambda(f_{i}))^{2}}{\sigma^{2}}\right)\right)^{\frac{-v+1}{2}}$

Parameters:	inv_link_f (Nx1 array) – latent variables link(f) y (Nx1 array) – data Y_metadata – Y_metadata which is not used in student t distribution
Returns:	likelihood evaluated for this point
Return type:	float

predictive_mean(mu, sigma, Y_metadata=None)[source]¶

predictive_variance(mu, variance, predictive_mean=None, Y_metadata=None)[source]¶

samples(gp, Y_metadata=None)[source]¶

Returns a set of samples of observations based on a given value of the latent variable.

Parameters:	gp – latent variable

update_gradients(grads)[source]¶: Pull out the gradients, be careful as the order must match the order in which the parameters are added

GPy.likelihoods package¶

Submodules¶

GPy.likelihoods.bernoulli module¶

GPy.likelihoods.exponential module¶

GPy.likelihoods.gamma module¶

GPy.likelihoods.gaussian module¶

GPy.likelihoods.likelihood module¶

GPy.likelihoods.link_functions module¶

GPy.likelihoods.mixed_noise module¶

GPy.likelihoods.poisson module¶

GPy.likelihoods.student_t module¶

Module contents¶

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