Model

model

Abstract Base classes that models should implement to be used for MCMC sampling.

The implementations of MCMC algorithms are based on (Cotter et al., 2013), We sample from a measure \(\mu\) that is defined via it's Radon-Nikodym derivative w.r.t. to a Gaussian reference measure.

\[ \frac{d\mu}{d\mu_0} \propto \exp(-\Phi(u)) \]

where \(\mu_0 = \mathcal{N}(0, C)\) is a centered Gaussian reference measure with covariance operator \(C\). The likelihood is then given via the potential \(\Phi(u)\). Models should at least implement calculating the potential \(\Phi(u)\) and the action (i.e. matrix vector product in the finite dimensional setting) of the covariance operator \(C\) on a vector. The covariace operator is also sometimes called preconditioner.

Classes:

Name	Description
MCMCModel	Abstract base class for an MCMC model that evaluates the likelihood
DifferentiableMCMCModel	Abstract base class for an MCMC model that evaluates the likelihood and also derivative information on the potential of the likelihood

References: Cotter, Roberts, Stuart, White (2013). MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster. Statistical Science 28(3).

ls_mcmc.model.MCMCModel

Bases: ABC

Base interface for MCMC target models.

Methods:

Name	Description
evaluate_potential	Compute potential of likelihood
compute_preconditioner_sqrt_action	Apply preconditioner on a state
reference_point	Reference point that can be used to make sampling more effective

reference_point abstractmethod property

reference_point: np.ndarray[tuple[int], np.dtype[np.floating]]

Return a reference point.

evaluate_potential abstractmethod

evaluate_potential(state: np.ndarray[tuple[int], np.dtype[np.floating]]) -> float

Return potential energy \(\Phi(u)\).

Parameters:

Name	Type	Description	Default
state	np.ndarray[tuple[int], np.dtype[np.floating]]	state \(u\).	required

compute_preconditioner_sqrt_action abstractmethod

compute_preconditioner_sqrt_action(state: np.ndarray[tuple[int], np.dtype[np.floating]]) -> np.ndarray[tuple[int], np.dtype[np.floating]]

Apply square-root preconditioner to state.

Parameters:

Name	Type	Description	Default
state	np.ndarray[tuple[int], np.dtype[np.floating]]	state \(u\).	required

ls_mcmc.model.DifferentiableMCMCModel

Bases: MCMCModel

Extension of MCMCModel with differentiable potential.

Methods:

Name	Description
evaluate_potential	Compute potential of likelihood
compute_preconditioner_sqrt_action	Apply square root of preconditioner on a state
reference_point	Reference point that can be used to make sampling more effective
evaluate_gradient_of_potential	Compute the gradient of the potential of the likelihood
compute_preconditioner_action	Apply preconditioner on a state

evaluate_gradient_of_potential abstractmethod

evaluate_gradient_of_potential(state: np.ndarray[tuple[int], np.dtype[np.floating]]) -> np.ndarray[tuple[int], np.dtype[np.floating]]

Return gradient \(\nabla\Phi(u)\).

Parameters:

Name	Type	Description	Default
state	np.ndarray[tuple[int], np.dtype[np.floating]]	state \(u\).	required

compute_preconditioner_action

compute_preconditioner_action(state: np.ndarray[tuple[int], np.dtype[np.floating]]) -> np.ndarray[tuple[int], np.dtype[np.floating]]

Apply full preconditioner.

Parameters:

Name	Type	Description	Default
state	np.ndarray[tuple[int], np.dtype[np.floating]]	state \(u\).	required