Integrators

gpjax.integrators

Likelihood = TypeVar('Likelihood', bound=Union['gpjax.likelihoods.AbstractLikelihood', None]) module-attribute

Gaussian = TypeVar('Gaussian', bound='gpjax.likelihoods.Gaussian') module-attribute

__all__ = ['AbstractIntegrator', 'GHQuadratureIntegrator', 'AnalyticalGaussianIntegrator'] module-attribute

AbstractIntegrator dataclass

Base class for integrators.

__init__() -> None

integrate(fun: Callable, y: Float[Array, 'N D'], mean: Float[Array, 'N D'], variance: Float[Array, 'N D'], likelihood: Likelihood) -> Float[Array, ' N'] abstractmethod

Integrate a function with respect to a Gaussian distribution.

Typically, the function will be the likelihood function and the mean and variance will be the parameters of the variational distribution.

Parameters:

Name	Type	Description	Default
fun	Callable	The function to be integrated.	required
y	Float[Array, 'N D']	The observed response variable.	required
mean	Float[Array, 'N D']	The mean of the variational distribution.	required
variance	Float[Array, 'N D']	The variance of the variational distribution.	required
likelihood	AbstractLikelihood	The likelihood function.	required

__call__(fun: Callable, y: Float[Array, 'N D'], mean: Float[Array, 'N D'], variance: Float[Array, 'N D'], likelihood: Likelihood) -> Float[Array, ' N']

Integrate a function with respect to a Gaussian distribution.

Typically, the function will be the likelihood function and the mean and variance will be the parameters of the variational distribution.

Parameters:

Name	Type	Description	Default
fun	Callable	The function to be integrated.	required
y	Float[Array, 'N D']	The observed response variable.	required
mean	Float[Array, 'N D']	The mean of the variational distribution.	required
variance	Float[Array, 'N D']	The variance of the variational distribution.	required
likelihood	AbstractLikelihood	The likelihood function.	required

GHQuadratureIntegrator dataclass

Bases: AbstractIntegrator

Compute an integral using Gauss-Hermite quadrature.

Gauss-Hermite quadrature is a method for approximating integrals through a weighted sum of function evaluations at specific points

$$\int F(t)\exp(-t^2)\mathrm{d}t \approx \sum_{j=1}^J w_j F(t_j)$$

where $t_j$ and $w_j$ are the roots and weights of the $J$-th order Hermite polynomial $H_J(t)$ that we can look up in table link.

num_points: int = 20 class-attribute instance-attribute

__call__(fun: Callable, y: Float[Array, 'N D'], mean: Float[Array, 'N D'], variance: Float[Array, 'N D'], likelihood: Likelihood) -> Float[Array, ' N']

Integrate a function with respect to a Gaussian distribution.

Typically, the function will be the likelihood function and the mean and variance will be the parameters of the variational distribution.

Parameters:

Name	Type	Description	Default
fun	Callable	The function to be integrated.	required
y	Float[Array, 'N D']	The observed response variable.	required
mean	Float[Array, 'N D']	The mean of the variational distribution.	required
variance	Float[Array, 'N D']	The variance of the variational distribution.	required
likelihood	AbstractLikelihood	The likelihood function.	required

__init__(num_points: int = 20) -> None

integrate(fun: Callable, y: Float[Array, 'N D'], mean: Float[Array, 'N D'], variance: Float[Array, 'N D'], likelihood: Likelihood) -> Float[Array, ' N']

Compute a quadrature integral.

Parameters:

Name	Type	Description	Default
fun	Callable	The likelihood to be integrated.	required
y	Float[Array, 'N D']	The observed response variable.	required
mean	Float[Array, 'N D']	The mean of the variational distribution.	required
variance	Float[Array, 'N D']	The variance of the variational distribution.	required
likelihood	AbstractLikelihood	The likelihood function.	required

Returns:

Type	Description
Float[Array, ' N']	Float[Array, 'N']: The expected log likelihood.

AnalyticalGaussianIntegrator dataclass

Bases: AbstractIntegrator

Compute the analytical integral of a Gaussian likelihood.

When the likelihood function is Gaussian, the integral can be computed in closed form. For a Gaussian likelihood $p(y|f) = \mathcal{N}(y|f, \sigma^2)$ and a variational distribution $q(f) = \mathcal{N}(f|m, s)$, the expected log-likelihood is given by

$$\mathbb{E}_{q(f)}[\log p(y|f)] = -\frac{1}{2}\left(\log(2\pi\sigma^2) + \frac{1}{\sigma^2}((y-m)^2 + s)\right)$$

__call__(fun: Callable, y: Float[Array, 'N D'], mean: Float[Array, 'N D'], variance: Float[Array, 'N D'], likelihood: Likelihood) -> Float[Array, ' N']

Integrate a function with respect to a Gaussian distribution.

Typically, the function will be the likelihood function and the mean and variance will be the parameters of the variational distribution.

Parameters:

Name	Type	Description	Default
fun	Callable	The function to be integrated.	required
y	Float[Array, 'N D']	The observed response variable.	required
mean	Float[Array, 'N D']	The mean of the variational distribution.	required
variance	Float[Array, 'N D']	The variance of the variational distribution.	required
likelihood	AbstractLikelihood	The likelihood function.	required

__init__() -> None

integrate(fun: Callable, y: Float[Array, 'N D'], mean: Float[Array, 'N D'], variance: Float[Array, 'N D'], likelihood: Gaussian) -> Float[Array, ' N']

Compute a Gaussian integral.

Parameters:

Name	Type	Description	Default
fun	Callable	The Gaussian likelihood to be integrated.	required
y	Float[Array, 'N D']	The observed response variable.	required
mean	Float[Array, 'N D']	The mean of the variational distribution.	required
variance	Float[Array, 'N D']	The variance of the variational distribution.	required
likelihood	Gaussian	The Gaussian likelihood function.	required

Returns:

Type	Description
Float[Array, ' N']	Float[Array, 'N']: The expected log likelihood.