Likelihoods

gpjax.likelihoods

tfb = tfp.bijectors module-attribute

tfd = tfp.distributions module-attribute

NonGaussian = Union[Poisson, Bernoulli] module-attribute

__all__ = ['AbstractLikelihood', 'NonGaussian', 'Gaussian', 'Bernoulli', 'Poisson', 'inv_probit'] module-attribute

AbstractLikelihood dataclass

Bases: Module

Abstract base class for likelihoods.

num_datapoints: int = static_field() class-attribute instance-attribute

integrator: AbstractIntegrator = static_field(GHQuadratureIntegrator()) class-attribute instance-attribute

__init_subclass__(mutable: bool = False)

replace(**kwargs: Any) -> Self

Replace the values of the fields of the object.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the fields of the object.	{}

Returns

Module: with the fields replaced.

replace_meta(**kwargs: Any) -> Self

Replace the metadata of the fields.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the metadata of the fields of the object.	{}

Returns

Module: with the metadata of the fields replaced.

update_meta(**kwargs: Any) -> Self

Update the metadata of the fields. The metadata must already exist.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the fields of the object.	{}

Returns

Module: with the fields replaced.

replace_trainable(**kwargs: Dict[str, bool]) -> Self

Replace the trainability status of local nodes of the Module.

replace_bijector(**kwargs: Dict[str, tfb.Bijector]) -> Self

Replace the bijectors of local nodes of the Module.

constrain() -> Self

Transform model parameters to the constrained space according to their defined bijectors.

Returns

Module: transformed to the constrained space.

unconstrain() -> Self

Transform model parameters to the unconstrained space according to their defined bijectors.

Returns

Module: transformed to the unconstrained space.

stop_gradient() -> Self

Stop gradients flowing through the Module.

Returns

Module: with gradients stopped.

trainables() -> Self

__init__(num_datapoints: int = static_field(), integrator: AbstractIntegrator = static_field(GHQuadratureIntegrator())) -> None

__call__(*args: Any, **kwargs: Any) -> tfd.Distribution

Evaluate the likelihood function at a given predictive distribution.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments to be passed to the likelihood's predict method.	()
**kwargs	Any	Keyword arguments to be passed to the likelihood's predict method.	{}

Returns

tfd.Distribution: The predictive distribution.

predict(*args: Any, **kwargs: Any) -> tfd.Distribution abstractmethod

Evaluate the likelihood function at a given predictive distribution.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments to be passed to the likelihood's predict method.	()
**kwargs	Any	Keyword arguments to be passed to the likelihood's predict method.	{}

Returns

tfd.Distribution: The predictive distribution.

link_function(f: Float[Array, ...]) -> tfd.Distribution abstractmethod

Return the link function of the likelihood function.

Returns

tfd.Distribution: The distribution of observations, y, given values of the
    Gaussian process, f.

expected_log_likelihood(y: Float[Array, 'N D'], mean: Float[Array, 'N D'], variance: Float[Array, 'N D']) -> Float[Array, ' N']

Compute the expected log likelihood.

For a variational distribution $q(f)\sim\mathcal{N}(m, s)$ and a likelihood $p(y|f)$, compute the expected log likelihood:

$$\mathbb{E}_{q(f)}\left[\log p(y|f)\right]$$

Parameters:

Name	Type	Description	Default
y	Float[Array, 'N D']	The observed response variable.	required
mean	Float[Array, 'N D']	The variational mean.	required
variance	Float[Array, 'N D']	The variational variance.	required

Returns:

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

Gaussian dataclass

Bases: AbstractLikelihood

Gaussian likelihood object.

Parameters:

Name	Type	Description	Default
obs_stddev	Union[ScalarFloat, Float[Array, '#N']]	the standard deviation of the Gaussian observation noise.	param_field(array(1.0), bijector=Softplus())

num_datapoints: int = static_field() class-attribute instance-attribute

obs_stddev: Union[ScalarFloat, Float[Array, '#N']] = param_field(jnp.array(1.0), bijector=tfb.Softplus()) class-attribute instance-attribute

integrator: AbstractIntegrator = static_field(AnalyticalGaussianIntegrator()) class-attribute instance-attribute

__init_subclass__(mutable: bool = False)

replace(**kwargs: Any) -> Self

Replace the values of the fields of the object.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the fields of the object.	{}

Returns

Module: with the fields replaced.

replace_meta(**kwargs: Any) -> Self

Replace the metadata of the fields.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the metadata of the fields of the object.	{}

Returns

Module: with the metadata of the fields replaced.

update_meta(**kwargs: Any) -> Self

Update the metadata of the fields. The metadata must already exist.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the fields of the object.	{}

Returns

Module: with the fields replaced.

replace_trainable(**kwargs: Dict[str, bool]) -> Self

Replace the trainability status of local nodes of the Module.

replace_bijector(**kwargs: Dict[str, tfb.Bijector]) -> Self

Replace the bijectors of local nodes of the Module.

constrain() -> Self

Transform model parameters to the constrained space according to their defined bijectors.

Returns

Module: transformed to the constrained space.

unconstrain() -> Self

Transform model parameters to the unconstrained space according to their defined bijectors.

Returns

Module: transformed to the unconstrained space.

stop_gradient() -> Self

Stop gradients flowing through the Module.

Returns

Module: with gradients stopped.

trainables() -> Self

__call__(*args: Any, **kwargs: Any) -> tfd.Distribution

Evaluate the likelihood function at a given predictive distribution.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments to be passed to the likelihood's predict method.	()
**kwargs	Any	Keyword arguments to be passed to the likelihood's predict method.	{}

Returns

tfd.Distribution: The predictive distribution.

expected_log_likelihood(y: Float[Array, 'N D'], mean: Float[Array, 'N D'], variance: Float[Array, 'N D']) -> Float[Array, ' N']

Compute the expected log likelihood.

For a variational distribution $q(f)\sim\mathcal{N}(m, s)$ and a likelihood $p(y|f)$, compute the expected log likelihood:

$$\mathbb{E}_{q(f)}\left[\log p(y|f)\right]$$

Parameters:

Name	Type	Description	Default
y	Float[Array, 'N D']	The observed response variable.	required
mean	Float[Array, 'N D']	The variational mean.	required
variance	Float[Array, 'N D']	The variational variance.	required

Returns:

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

__init__(num_datapoints: int = static_field(), integrator: AbstractIntegrator = static_field(AnalyticalGaussianIntegrator()), obs_stddev: Union[ScalarFloat, Float[Array, '#N']] = param_field(jnp.array(1.0), bijector=tfb.Softplus())) -> None

link_function(f: Float[Array, ...]) -> tfd.Normal

The link function of the Gaussian likelihood.

Parameters:

Name	Type	Description	Default
f	Float[Array, ...]	Function values.	required

Returns

tfd.Normal: The likelihood function.

predict(dist: Union[tfd.MultivariateNormalTriL, GaussianDistribution]) -> tfd.MultivariateNormalFullCovariance

Evaluate the Gaussian likelihood.

Evaluate the Gaussian likelihood function at a given predictive distribution. Computationally, this is equivalent to summing the observation noise term to the diagonal elements of the predictive distribution's covariance matrix.

Parameters:

Name	Type	Description	Default
dist	Distribution	The Gaussian process posterior, evaluated at a finite set of test points.	required

Returns

tfd.Distribution: The predictive distribution.

Bernoulli dataclass

Bases: AbstractLikelihood

num_datapoints: int = static_field() class-attribute instance-attribute

integrator: AbstractIntegrator = static_field(GHQuadratureIntegrator()) class-attribute instance-attribute

__init_subclass__(mutable: bool = False)

replace(**kwargs: Any) -> Self

Replace the values of the fields of the object.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the fields of the object.	{}

Returns

Module: with the fields replaced.

replace_meta(**kwargs: Any) -> Self

Replace the metadata of the fields.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the metadata of the fields of the object.	{}

Returns

Module: with the metadata of the fields replaced.

update_meta(**kwargs: Any) -> Self

Update the metadata of the fields. The metadata must already exist.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the fields of the object.	{}

Returns

Module: with the fields replaced.

replace_trainable(**kwargs: Dict[str, bool]) -> Self

Replace the trainability status of local nodes of the Module.

replace_bijector(**kwargs: Dict[str, tfb.Bijector]) -> Self

Replace the bijectors of local nodes of the Module.

constrain() -> Self

Transform model parameters to the constrained space according to their defined bijectors.

Returns

Module: transformed to the constrained space.

unconstrain() -> Self

Transform model parameters to the unconstrained space according to their defined bijectors.

Returns

Module: transformed to the unconstrained space.

stop_gradient() -> Self

Stop gradients flowing through the Module.

Returns

Module: with gradients stopped.

trainables() -> Self

__call__(*args: Any, **kwargs: Any) -> tfd.Distribution

Evaluate the likelihood function at a given predictive distribution.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments to be passed to the likelihood's predict method.	()
**kwargs	Any	Keyword arguments to be passed to the likelihood's predict method.	{}

Returns

tfd.Distribution: The predictive distribution.

expected_log_likelihood(y: Float[Array, 'N D'], mean: Float[Array, 'N D'], variance: Float[Array, 'N D']) -> Float[Array, ' N']

Compute the expected log likelihood.

For a variational distribution $q(f)\sim\mathcal{N}(m, s)$ and a likelihood $p(y|f)$, compute the expected log likelihood:

$$\mathbb{E}_{q(f)}\left[\log p(y|f)\right]$$

Parameters:

Name	Type	Description	Default
y	Float[Array, 'N D']	The observed response variable.	required
mean	Float[Array, 'N D']	The variational mean.	required
variance	Float[Array, 'N D']	The variational variance.	required

Returns:

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

__init__(num_datapoints: int = static_field(), integrator: AbstractIntegrator = static_field(GHQuadratureIntegrator())) -> None

link_function(f: Float[Array, ...]) -> tfd.Distribution

The probit link function of the Bernoulli likelihood.

Parameters:

Name	Type	Description	Default
f	Float[Array, ...]	Function values.	required

Returns

tfd.Distribution: The likelihood function.

predict(dist: tfd.Distribution) -> tfd.Distribution

Evaluate the pointwise predictive distribution.

Evaluate the pointwise predictive distribution, given a Gaussian process posterior and likelihood parameters.

Parameters:

Name	Type	Description	Default
dist	Distribution	The Gaussian process posterior, evaluated at a finite set of test points.	required

Returns

tfd.Distribution: The pointwise predictive distribution.

Poisson dataclass

Bases: AbstractLikelihood

num_datapoints: int = static_field() class-attribute instance-attribute

integrator: AbstractIntegrator = static_field(GHQuadratureIntegrator()) class-attribute instance-attribute

__init_subclass__(mutable: bool = False)

replace(**kwargs: Any) -> Self

Replace the values of the fields of the object.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the fields of the object.	{}

Returns

Module: with the fields replaced.

replace_meta(**kwargs: Any) -> Self

Replace the metadata of the fields.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the metadata of the fields of the object.	{}

Returns

Module: with the metadata of the fields replaced.

update_meta(**kwargs: Any) -> Self

Update the metadata of the fields. The metadata must already exist.

Parameters:

Name	Type	Description	Default
**kwargs	Any	keyword arguments to replace the fields of the object.	{}

Returns

Module: with the fields replaced.

replace_trainable(**kwargs: Dict[str, bool]) -> Self

Replace the trainability status of local nodes of the Module.

replace_bijector(**kwargs: Dict[str, tfb.Bijector]) -> Self

Replace the bijectors of local nodes of the Module.

constrain() -> Self

Transform model parameters to the constrained space according to their defined bijectors.

Returns

Module: transformed to the constrained space.

unconstrain() -> Self

Transform model parameters to the unconstrained space according to their defined bijectors.

Returns

Module: transformed to the unconstrained space.

stop_gradient() -> Self

Stop gradients flowing through the Module.

Returns

Module: with gradients stopped.

trainables() -> Self

__call__(*args: Any, **kwargs: Any) -> tfd.Distribution

Evaluate the likelihood function at a given predictive distribution.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments to be passed to the likelihood's predict method.	()
**kwargs	Any	Keyword arguments to be passed to the likelihood's predict method.	{}

Returns

tfd.Distribution: The predictive distribution.

expected_log_likelihood(y: Float[Array, 'N D'], mean: Float[Array, 'N D'], variance: Float[Array, 'N D']) -> Float[Array, ' N']

Compute the expected log likelihood.

For a variational distribution $q(f)\sim\mathcal{N}(m, s)$ and a likelihood $p(y|f)$, compute the expected log likelihood:

$$\mathbb{E}_{q(f)}\left[\log p(y|f)\right]$$

Parameters:

Name	Type	Description	Default
y	Float[Array, 'N D']	The observed response variable.	required
mean	Float[Array, 'N D']	The variational mean.	required
variance	Float[Array, 'N D']	The variational variance.	required

Returns:

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

__init__(num_datapoints: int = static_field(), integrator: AbstractIntegrator = static_field(GHQuadratureIntegrator())) -> None

link_function(f: Float[Array, ...]) -> tfd.Distribution

The link function of the Poisson likelihood.

Parameters:

Name	Type	Description	Default
f	Float[Array, ...]	Function values.	required

Returns:

Type	Description
Distribution	tfd.Distribution: The likelihood function.

predict(dist: tfd.Distribution) -> tfd.Distribution

Evaluate the pointwise predictive distribution.

Evaluate the pointwise predictive distribution, given a Gaussian process posterior and likelihood parameters.

Parameters:

Name	Type	Description	Default
dist	Distribution	The Gaussian process posterior, evaluated at a finite set of test points.	required

Returns:

Type	Description
Distribution	tfd.Distribution: The pointwise predictive distribution.

inv_probit(x: Float[Array, ' *N']) -> Float[Array, ' *N']

Compute the inverse probit function.

Parameters:

Name	Type	Description	Default
x	Float[Array, '*N']	A vector of values.	required

Returns

Float[Array, "*N"]: The inverse probit of the input vector.