Prior Object

prior

Prior distribution interface.

Class

Prior: Prior distribution class.

ls_prior.prior.Prior

Prior distribution class.

This class implements functionality of a Gaussian prior measure that is typically required in the context of Bayesian inference. In the spirit of separation of concerns, the prior component is rather stupid. It simply utilizes objects for the (representation of) a covariance operator \(\mathcal{C}\), its factorization \(\widehat{\mathcal{C}}\), and a precision operator \(\mathcal{C}^{-1}\). These components have to adhere to the interface prescribed by the InterfaceComponent class. The necessary objects can be manually assembled and be handed to the Prior class for maximum flexibility. On the other hand, the builder module provides a convenient alternative for the setup of a preconfigured prior object.

Methods:

Name	Description
evaluate_cost	Evaluate the cost/negative log-probability for a given parameter vector.
evaluate_gradient	Evaluate the gradient of the cost functional with respect to a given parameter vector.
evaluate_hessian_vector_product	Evaluate the Hessian-vector product of the cost functional in the given direction.
sample	Generate a sample from the prior distribution.

random_vector_size property

random_vector_size: int

Return the required size of the random vector for sampling.

__init__

__init__(
    mean_vector: np.ndarray[tuple[int], np.dtype[np.float64]],
    precision_operator: components.InterfaceComponent,
    covariance_operator: components.InterfaceComponent,
    covariance_factorization: components.InterfaceComponent,
    fem_converter: fem.FEMConverter,
    seed: int,
) -> None

Initialize Prior distribution object.

Parameters:

Name	Type	Description	Default
mean_vector	np.ndarray[tuple[int], np.dtype[np.float64]]	Mean vector \(\overline{m}\) of the prior measure.	required
precision_operator	components.InterfaceComponent	Representation of the precision operator \(\mathcal{C}^{-1}\).	required
covariance_operator	components.InterfaceComponent	Representation of the covariance operator \(\mathcal{C}\).	required
covariance_factorization	components.InterfaceComponent	Representation of a factorization \(\widehat{\mathcal{C}}\) of the covariance operator for sampling.	required
fem_converter	fem.FEMConverter	Converter object to switch between vertex- and DoF-based representation of vectors.	required
seed	int	Random seed for the internal random number generator.	required

Raises:

Type	Description
ValueError	Checks that precision operator has the correct shape.
ValueError	Checks that covariance operator has the correct shape.
ValueError	Checks that covariance factor has the correct shape.

evaluate_cost

evaluate_cost(parameter_vector: np.ndarray[tuple[int], np.dtype[np.float64]]) -> float

Evaluate the cost functional, i.e. the negative log probability for given input.

Computes \(\frac{1}{2} (m-\overline{m})^T \mathcal{C}^{-1} (m-\overline{m})\).

Parameters:

Name	Type	Description	Default
parameter_vector	np.ndarray[tuple[int], np.dtype[np.float64]]	Parameter candidate for which to evaluate the cost/negative log probability, given on mesh vertices.	required

Returns:

Name	Type	Description
float	float	cost/negative log probability

evaluate_gradient

evaluate_gradient(
    parameter_vector: np.ndarray[tuple[int], np.dtype[np.float64]],
) -> np.ndarray[tuple[int], np.dtype[np.float64]]

Evaluate the gradient of the cost/negative log-probability w.r.t. the parameter vector.

Computes \(\mathcal{C}^{-1} (m - \overline{m})\).

Parameters:

Name	Type	Description	Default
parameter_vector	np.ndarray[tuple[int], np.dtype[np.float64]]	Parameter candidate for which to evaluate the gradient, given on mesh vertices.	required

Returns:

Type	Description
np.ndarray[tuple[int], np.dtype[np.float64]]	np.ndarray[tuple[int], np.dtype[np.float64]]: Gradient of the cost/negative log-probability, given on mesh vertices.

evaluate_hessian_vector_product

evaluate_hessian_vector_product(
    direction_vector: np.ndarray[tuple[int], np.dtype[np.float64]],
) -> np.ndarray[tuple[int], np.dtype[np.float64]]

Evaluate the application of the Hessian of the cost functional on a given direction.

Computes \(\mathcal{C}^{-1} m_{\text{dir}}\).

Parameters:

Name	Type	Description	Default
direction_vector	np.ndarray[tuple[int], np.dtype[np.float64]]	Direction for which to evaluate the Hessian-vector product, given on mesh vertices.	required

Returns:

Type	Description
np.ndarray[tuple[int], np.dtype[np.float64]]	np.ndarray[tuple[int], np.dtype[np.float64]]: Hessian-vector product, given on mesh vertices.

generate_sample

generate_sample() -> np.ndarray[tuple[int], np.dtype[np.float64]]

Generate a sample from the prior measure.

Computes sample in two-step procedure:

1. Generate i.i.d. normal vector \(\xi\) matching input dimension of \(\widehat{\mathcal{C}}\).
2. Multiply with covariance factorization \(\widehat{\mathcal{C}}\) and add mean

Returns:

Type	Description
np.ndarray[tuple[int], np.dtype[np.float64]]	np.ndarray[tuple[int], np.dtype[np.float64]]: Sample vector, given on mesh vertices.

apply_covariance_operator

apply_covariance_operator(
    parameter_vector: np.ndarray[tuple[int], np.dtype[np.float64]],
) -> np.ndarray[tuple[int], np.dtype[np.float64]]

Apply the covariance operator to a given parameter vector.

Parameters:

Name	Type	Description	Default
parameter_vector	np.ndarray[tuple[int], np.dtype[np.float64]]	Parameter candidate for which to apply the covariance operator, given on mesh vertices.	required

Returns:

Type	Description
np.ndarray[tuple[int], np.dtype[np.float64]]	np.ndarray[tuple[int], np.dtype[np.float64]]: Result of applying the covariance operator, given on mesh vertices.

apply_covariance_factorization

apply_covariance_factorization(
    random_vector: np.ndarray[tuple[int], np.dtype[np.float64]],
) -> np.ndarray[tuple[int], np.dtype[np.float64]]

Apply the covariance factorization to a given parameter vector.

Parameters:

Name	Type	Description	Default
random_vector	np.ndarray[tuple[int], np.dtype[np.float64]]	Random vector for which to apply the covariance factorization.	required

Returns:

Type	Description
np.ndarray[tuple[int], np.dtype[np.float64]]	np.ndarray[tuple[int], np.dtype[np.float64]]: Result of applying the covariance factorization, given on mesh vertices.

Raises:

Type	Description
ValueError	Checks that the random vector has the correct shape.

apply_precision_operator

apply_precision_operator(
    parameter_vector: np.ndarray[tuple[int], np.dtype[np.float64]],
) -> np.ndarray[tuple[int], np.dtype[np.float64]]

Apply the precision operator to a given parameter vector.

Parameters:

Name	Type	Description	Default
parameter_vector	np.ndarray[tuple[int], np.dtype[np.float64]]	Parameter candidate for which to apply the precision operator, given on mesh vertices.	required

Returns:

Type	Description
np.ndarray[tuple[int], np.dtype[np.float64]]	np.ndarray[tuple[int], np.dtype[np.float64]]: Result of applying the precision operator, given on mesh vertices.

_check_input_dimension

_check_input_dimension(vector: np.ndarray) -> None

Checks dimension of input vectors, applied in all interface methods.

_check_output_dimension

_check_output_dimension(vector: np.ndarray) -> None

Checks dimension of result vectors, applied in all interface methods.