Laplace Approximation of the Posterior¶

laplace ¶

Wrapper for the Gaussian low-rank posterior object in Hippylib.

This module provides the functionality for the so-called Laplace approximation of the posterior, a method of variational inference. The Laplace approximation is based on the linearization of the forward map that governs the inverse problem, typically about the MAP point. In combination with a Gaussian prior and likelihood, the Laplace approximation results in a Gaussian posterior, defined by a mean function (the MAP estimate), and the local Hessian at that point.

The Laplace-approximation implemented in SPINrelies on the GaussianLRPosterior object in Hippylib, which takes a low-rank approximation of the Hessian as input. This low-rank approximation is computed in SPIN using the compute_low_rank_hessian function. The MAP can be found using the Newton-CG solver in Hippylib, which is wrapped in the NewtonCGSolver class for SPIN applications.

Classes:

Name	Description
`LowRankLaplaceApproximationSettings`	Input for the low-rank Laplace approximation object.
`LowRankLaplaceApproximation`	Wrapper for the low-rank Laplace approximation object.

spin.hippylib.laplace.LowRankLaplaceApproximationSettings `dataclass` ¶

Input for the low-rank Laplace approximation object.

Attributes:

Name	Type	Description
`inference_model`	`hl.Model`	Hippylib inference model object.
`mean`	`npt.NDArray[np.floating]`	Mean function (usually the MAP).
`low_rank_hessian_eigenvalues`	`npt.NDArray[np.floating]`	Eigenvalues for low-rank approximation of the negative log-posterior Hessian (at the MAP).
`low_rank_hessian_eigenvectors`	`Iterable[npt.NDArray[np.floating]]`	Eigenvectors for low-rank approximation of the negative log-posterior Hessian (at the MAP).

spin.hippylib.laplace.LowRankLaplaceApproximation ¶

Low-rank Laplace approximation of the posterior distribution.

This class provides a wrapper for the GaussianLRPosterior object in Hippylib with a more modern and user-friendly interface.

Attributes:

Name	Type	Description
`hippylib_gaussian_posterior`	`hl.GaussianLRPosterior`	Underlying Hippylib object

Methods:

Name	Description
`compute_pointwise_variance`	Compute the pointwise variance for the Laplace approximation.

hippylib_gaussian_posterior `property` ¶

hippylib_gaussian_posterior: hl.GaussianLRPosterior

Return the underlying GaussianLRPosterior Hippylib object.

init ¶

__init__(settings: LowRankLaplaceApproximationSettings) -> None

Initialize the underlying hippylib object.

Parameters:

Name	Type	Description	Default
`settings`	`LowRankLaplaceApproximationSettings`	Input data for the approximation.	required

compute_pointwise_variance ¶

compute_pointwise_variance(method: Annotated[str, Is[lambda x: x in Exact, Randomized]], num_eigenvalues_randomized: Annotated[int, Is[lambda x: x > 0]] | None = None) -> npt.NDArray[np.floating]

Compute the pointwise variance field of the Laplace approximation.

For a detailed description of the different methods, we refer to the documentation of the GaussianLRPosterior object in Hippylib.

Parameters:

Name	Type	Description	Default
`method`	`str`	Algorithm for computation of the variance, options are "`Exact`", and "`Randomized`".	required
`num_eigenvalues_randomized`	`int`	Number of dominant eigenvalues for the randomized algorithm. Defaults to None.	`None`

Raises:

Type	Description
`ValueError`	Checks that `num_eigenvalues_randomized` is provided for the 'Randomized' algorithm

Returns:

Type	Description
`npt.NDArray[np.floating]`	npt.NDArray[np.floating]: Pointwise variance field.