Variational Families

gpjax.variational_families

__all__ = ['AbstractVariationalFamily', 'AbstractVariationalGaussian', 'VariationalGaussian', 'WhitenedVariationalGaussian', 'NaturalVariationalGaussian', 'ExpectationVariationalGaussian', 'CollapsedVariationalGaussian'] module-attribute

AbstractVariationalFamily dataclass

Bases: Module

Abstract base class used to represent families of distributions that can be used within variational inference.

posterior: AbstractPosterior 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__(posterior: AbstractPosterior) -> None

__call__(*args: Any, **kwargs: Any) -> GaussianDistribution

Evaluate the variational family's density.

For a given set of parameters, compute the latent function's prediction under the variational approximation.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments of the variational family's predict method.	()
**kwargs	Any	Keyword arguments of the variational family's predict method.	{}

Returns

GaussianDistribution: The output of the variational family's `predict` method.

predict(*args: Any, **kwargs: Any) -> GaussianDistribution abstractmethod

Predict the GP's output given the input.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments of the variational family's predict method.	()
**kwargs	Any	Keyword arguments of the variational family's predict method.	{}

Returns

GaussianDistribution: The output of the variational family's ``predict`` method.

AbstractVariationalGaussian dataclass

Bases: AbstractVariationalFamily

The variational Gaussian family of probability distributions.

posterior: AbstractPosterior instance-attribute

inducing_inputs: Float[Array, 'N D'] instance-attribute

jitter: ScalarFloat = static_field(1e-06) class-attribute instance-attribute

num_inducing: int property

The number of inducing inputs.

__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) -> GaussianDistribution

Evaluate the variational family's density.

For a given set of parameters, compute the latent function's prediction under the variational approximation.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments of the variational family's predict method.	()
**kwargs	Any	Keyword arguments of the variational family's predict method.	{}

Returns

GaussianDistribution: The output of the variational family's `predict` method.

predict(*args: Any, **kwargs: Any) -> GaussianDistribution abstractmethod

Predict the GP's output given the input.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments of the variational family's predict method.	()
**kwargs	Any	Keyword arguments of the variational family's predict method.	{}

Returns

GaussianDistribution: The output of the variational family's ``predict`` method.

__init__(posterior: AbstractPosterior, inducing_inputs: Float[Array, 'N D'], jitter: ScalarFloat = static_field(1e-06)) -> None

VariationalGaussian dataclass

Bases: AbstractVariationalGaussian

The variational Gaussian family of probability distributions.

The variational family is $q(f(\cdot)) = \int p(f(\cdot)\mid u) q(u) \mathrm{d}u$, where $u = f(z)$ are the function values at the inducing inputs $z$ and the distribution over the inducing inputs is $q(u) = \mathcal{N}(\mu, S)$. We parameterise this over $\mu$ and $sqrt$ with $S = sqrt sqrt^{\top}$.

posterior: AbstractPosterior instance-attribute

inducing_inputs: Float[Array, 'N D'] instance-attribute

jitter: ScalarFloat = static_field(1e-06) class-attribute instance-attribute

num_inducing: int property

The number of inducing inputs.

variational_mean: Union[Float[Array, 'N 1'], None] = param_field(None) class-attribute instance-attribute

variational_root_covariance: Float[Array, 'N N'] = param_field(None, bijector=tfb.FillTriangular()) 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) -> GaussianDistribution

Evaluate the variational family's density.

For a given set of parameters, compute the latent function's prediction under the variational approximation.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments of the variational family's predict method.	()
**kwargs	Any	Keyword arguments of the variational family's predict method.	{}

Returns

GaussianDistribution: The output of the variational family's `predict` method.

__init__(posterior: AbstractPosterior, inducing_inputs: Float[Array, 'N D'], jitter: ScalarFloat = static_field(1e-06), variational_mean: Union[Float[Array, 'N 1'], None] = param_field(None), variational_root_covariance: Float[Array, 'N N'] = param_field(None, bijector=tfb.FillTriangular())) -> None

__post_init__() -> None

prior_kl() -> ScalarFloat

Compute the prior KL divergence.

Compute the KL-divergence between our variational approximation and the Gaussian process prior.

For this variational family, we have

$$\begin{align} \operatorname{KL}[q(f(\cdot))\mid\mid p(\cdot)] & = \operatorname{KL}[q(u)\mid\mid p(u)]\\ & = \operatorname{KL}[ \mathcal{N}(\mu, S) \mid\mid N(\mu z, \mathbf{K}_{zz}) ], \end{align}$$

where $u = f(z)$ and $z$ are the inducing inputs.

Returns

 ScalarFloat: The KL-divergence between our variational
    approximation and the GP prior.

predict(test_inputs: Float[Array, 'N D']) -> GaussianDistribution

Compute the predictive distribution of the GP at the test inputs t.

This is the integral $q(f(t)) = \int p(f(t)\mid u) q(u) \mathrm{d}u$, which can be computed in closed form as:

$$\mathcal{N}\left(f(t); \mu t + \mathbf{K}_{tz} \mathbf{K}_{zz}^{-1} (\mu - \mu z), \mathbf{K}_{tt} - \mathbf{K}_{tz} \mathbf{K}_{zz}^{-1} \mathbf{K}_{zt} + \mathbf{K}_{tz} \mathbf{K}_{zz}^{-1} S \mathbf{K}_{zz}^{-1} \mathbf{K}_{zt}\right).$$

Parameters:

Name	Type	Description	Default
test_inputs	Float[Array, 'N D']	The test inputs at which we wish to make a prediction.	required

Returns

GaussianDistribution: The predictive distribution of the low-rank GP at
    the test inputs.

WhitenedVariationalGaussian dataclass

Bases: VariationalGaussian

The whitened variational Gaussian family of probability distributions.

The variational family is $q(f(\cdot)) = \int p(f(\cdot)\mid u) q(u) \mathrm{d}u$, where $u = f(z)$ are the function values at the inducing inputs $z$ and the distribution over the inducing inputs is $q(u) = \mathcal{N}(Lz \mu + mz, Lz S Lz^{\top})$. We parameterise this over $\mu$ and $sqrt$ with $S = sqrt sqrt^{\top}$.

posterior: AbstractPosterior instance-attribute

inducing_inputs: Float[Array, 'N D'] instance-attribute

jitter: ScalarFloat = static_field(1e-06) class-attribute instance-attribute

num_inducing: int property

The number of inducing inputs.

variational_mean: Union[Float[Array, 'N 1'], None] = param_field(None) class-attribute instance-attribute

variational_root_covariance: Float[Array, 'N N'] = param_field(None, bijector=tfb.FillTriangular()) 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) -> GaussianDistribution

Evaluate the variational family's density.

For a given set of parameters, compute the latent function's prediction under the variational approximation.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments of the variational family's predict method.	()
**kwargs	Any	Keyword arguments of the variational family's predict method.	{}

Returns

GaussianDistribution: The output of the variational family's `predict` method.

__post_init__() -> None

__init__(posterior: AbstractPosterior, inducing_inputs: Float[Array, 'N D'], jitter: ScalarFloat = static_field(1e-06), variational_mean: Union[Float[Array, 'N 1'], None] = param_field(None), variational_root_covariance: Float[Array, 'N N'] = param_field(None, bijector=tfb.FillTriangular())) -> None

prior_kl() -> ScalarFloat

Compute the KL-divergence between our variational approximation and the Gaussian process prior.

For this variational family, we have

$$\begin{align} \operatorname{KL}[q(f(\cdot))\mid\mid p(\cdot)] & = \operatorname{KL}[q(u)\mid\mid p(u)]\\ & = \operatorname{KL}[N(\mu , S)\mid\mid N(0, I)]. \end{align}$$

Returns

ScalarFloat: The KL-divergence between our variational
    approximation and the GP prior.

predict(test_inputs: Float[Array, 'N D']) -> GaussianDistribution

Compute the predictive distribution of the GP at the test inputs t.

This is the integral q(f(t)) = \int p(f(t)\midu) q(u) du, which can be computed in closed form as

$$\mathcal{N}\left(f(t); \mu t + \mathbf{K}_{tz} \mathbf{L}z^{\top} \mu , \mathbf{K}_{tt} - \mathbf{K}_{tz} \mathbf{K}_{zz}^{-1} \mathbf{K}_{zt} + \mathbf{K}_{tz} \mathbf{L}z^{\top} S \mathbf{L}z^{-1} \mathbf{K}_{zt} \right).$$

Parameters:

Name	Type	Description	Default
test_inputs	Float[Array, 'N D']	The test inputs at which we wish to make a prediction.	required

Returns

GaussianDistribution: The predictive distribution of the low-rank GP at
    the test inputs.

NaturalVariationalGaussian dataclass

Bases: AbstractVariationalGaussian

The natural variational Gaussian family of probability distributions.

The variational family is $q(f(\cdot)) = \int p(f(\cdot)\mid u) q(u) \mathrm{d}u$, where $u = f(z)$ are the function values at the inducing inputs $z$ and the distribution over the inducing inputs is $q(u) = N(\mu, S)$. Expressing the variational distribution, in the form of the exponential family, $q(u) = exp(\theta^{\top} T(u) - a(\theta))$, gives rise to the natural parameterisation $\theta = (\theta_{1}, \theta_{2}) = (S^{-1}\mu, -S^{-1}/2)$, to perform model inference, where $T(u) = [u, uu^{\top}]$ are the sufficient statistics.

posterior: AbstractPosterior instance-attribute

inducing_inputs: Float[Array, 'N D'] instance-attribute

jitter: ScalarFloat = static_field(1e-06) class-attribute instance-attribute

num_inducing: int property

The number of inducing inputs.

natural_vector: Float[Array, 'M 1'] = None class-attribute instance-attribute

natural_matrix: Float[Array, 'M M'] = None 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) -> GaussianDistribution

Evaluate the variational family's density.

For a given set of parameters, compute the latent function's prediction under the variational approximation.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments of the variational family's predict method.	()
**kwargs	Any	Keyword arguments of the variational family's predict method.	{}

Returns

GaussianDistribution: The output of the variational family's `predict` method.

__init__(posterior: AbstractPosterior, inducing_inputs: Float[Array, 'N D'], jitter: ScalarFloat = static_field(1e-06), natural_vector: Float[Array, 'M 1'] = None, natural_matrix: Float[Array, 'M M'] = None) -> None

__post_init__()

prior_kl() -> ScalarFloat

Compute the KL-divergence between our current variational approximation and the Gaussian process prior.

For this variational family, we have

$$\begin{align} \operatorname{KL}[q(f(\cdot))\mid\mid p(\cdot)] & = \operatorname{KL}[q(u)\mid\mid p(u)] \\ & = \operatorname{KL}[N(\mu, S)\mid\mid N(mz, \mathbf{K}_{zz})], \end{align}$$

with $\mu$ and $S$ computed from the natural parameterisation $\theta = (S^{-1}\mu , -S^{-1}/2)$.

Returns

ScalarFloat: The KL-divergence between our variational approximation and
    the GP prior.

predict(test_inputs: Float[Array, 'N D']) -> GaussianDistribution

Compute the predictive distribution of the GP at the test inputs $t$.

This is the integral $q(f(t)) = \int p(f(t)\mid u) q(u) \mathrm{d}u$, which can be computed in closed form as

$$\mathcal{N}\left(f(t); \mu t + \mathbf{K}_{tz} \mathbf{K}_{zz}^{-1} (\mu - \mu z), \mathbf{K}_{tt} - \mathbf{K}_{tz} \mathbf{K}_{zz}^{-1} \mathbf{K}_{zt} + \mathbf{K}_{tz} \mathbf{K}_{zz}^{-1} S \mathbf{K}_{zz}^{-1} \mathbf{K}_{zt} \right),$$

with $\mu$ and $S$ computed from the natural parameterisation $\theta = (S^{-1}\mu , -S^{-1}/2)$.

Returns

GaussianDistribution: A function that accepts a set of test points and will
    return the predictive distribution at those points.

ExpectationVariationalGaussian dataclass

Bases: AbstractVariationalGaussian

The natural variational Gaussian family of probability distributions.

The variational family is $q(f(\cdot)) = \int p(f(\cdot)\mid u) q(u) \mathrm{d}u$, where $u = f(z)$ are the function values at the inducing inputs $z$ and the distribution over the inducing inputs is $q(u) = \mathcal{N}(\mu, S)$. Expressing the variational distribution, in the form of the exponential family, $q(u) = exp(\theta^{\top} T(u) - a(\theta))$, gives rise to the natural parameterisation $\theta = (\theta_{1}, \theta_{2}) = (S^{-1}\mu , -S^{-1}/2)$ and sufficient statistics $T(u) = [u, uu^{\top}]$. The expectation parameters are given by $\nu = \int T(u) q(u) \mathrm{d}u$. This gives a parameterisation, $\nu = (\nu_{1}, \nu_{2}) = (\mu , S + uu^{\top})$ to perform model inference over.

posterior: AbstractPosterior instance-attribute

inducing_inputs: Float[Array, 'N D'] instance-attribute

jitter: ScalarFloat = static_field(1e-06) class-attribute instance-attribute

num_inducing: int property

The number of inducing inputs.

expectation_vector: Float[Array, 'M 1'] = None class-attribute instance-attribute

expectation_matrix: Float[Array, 'M M'] = None 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) -> GaussianDistribution

Evaluate the variational family's density.

For a given set of parameters, compute the latent function's prediction under the variational approximation.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments of the variational family's predict method.	()
**kwargs	Any	Keyword arguments of the variational family's predict method.	{}

Returns

GaussianDistribution: The output of the variational family's `predict` method.

__init__(posterior: AbstractPosterior, inducing_inputs: Float[Array, 'N D'], jitter: ScalarFloat = static_field(1e-06), expectation_vector: Float[Array, 'M 1'] = None, expectation_matrix: Float[Array, 'M M'] = None) -> None

__post_init__()

prior_kl() -> ScalarFloat

Evaluate the prior KL-divergence.

Compute the KL-divergence between our current variational approximation and the Gaussian process prior.

For this variational family, we have

$$\begin{align} \operatorname{KL}(q(f(\cdot))\mid\mid p(\cdot)) & = \operatorname{KL}(q(u)\mid\mid p(u)) \\ & =\operatorname{KL}(\mathcal{N}(\mu, S)\mid\mid \mathcal{N}(m_z, K_{zz})), \end{align}$$

where $\mu$ and $S$ are the expectation parameters of the variational distribution and $m_z$ and $K_{zz}$ are the mean and covariance of the prior distribution.

Returns

ScalarFloat: The KL-divergence between our variational approximation and
    the GP prior.

predict(test_inputs: Float[Array, 'N D']) -> GaussianDistribution

Evaluate the predictive distribution.

Compute the predictive distribution of the GP at the test inputs $t$.

This is the integral $q(f(t)) = \int p(f(t)\mid u)q(u)\mathrm{d}u$, which can be computed in closed form as which can be computed in closed form as

$$\mathcal{N}(f(t); \mu_t + \mathbf{K}_{tz}\mathbf{K}_{zz}^{-1}(\mu - \mu_z), \mathbf{K}_{tt} - \mathbf{K}_{tz}\mathbf{K}_{zz}^{-1}\mathbf{K}_{zt} + \mathbf{K}_{tz}\mathbf{K}_{zz}^{-1}\mathbf{S} \mathbf{K}_{zz}^{-1}\mathbf{K}_{zt})$$

with $\mu$ and $S$ computed from the expectation parameterisation $\eta = (\mu, S + uu^\top)$.

Returns

GaussianDistribution: The predictive distribution of the GP at the
    test inputs $t$.

CollapsedVariationalGaussian dataclass

Bases: AbstractVariationalGaussian

Collapsed variational Gaussian.

Collapsed variational Gaussian family of probability distributions. The key reference is Titsias, (2009) - Variational Learning of Inducing Variables in Sparse Gaussian Processes.

posterior: AbstractPosterior instance-attribute

inducing_inputs: Float[Array, 'N D'] instance-attribute

jitter: ScalarFloat = static_field(1e-06) class-attribute instance-attribute

num_inducing: int property

The number of inducing inputs.

__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) -> GaussianDistribution

Evaluate the variational family's density.

For a given set of parameters, compute the latent function's prediction under the variational approximation.

Parameters:

Name	Type	Description	Default
*args	Any	Arguments of the variational family's predict method.	()
**kwargs	Any	Keyword arguments of the variational family's predict method.	{}

Returns

GaussianDistribution: The output of the variational family's `predict` method.

__init__(posterior: AbstractPosterior, inducing_inputs: Float[Array, 'N D'], jitter: ScalarFloat = static_field(1e-06)) -> None

__post_init__()

predict(test_inputs: Float[Array, 'N D'], train_data: Dataset) -> GaussianDistribution

Compute the predictive distribution of the GP at the test inputs.

Parameters:

Name	Type	Description	Default
test_inputs	Float[Array, 'N D']	The test inputs $t$ at which to make predictions.	required
train_data	Dataset	The training data that was used to fit the GP.	required

Returns

GaussianDistribution: The predictive distribution of the collapsed
    variational Gaussian process at the test inputs $t$.