BayesianQuadrature

class probnum.quad.solvers.BayesianQuadrature(kernel, measure, policy, belief_update, stopping_criterion, initial_design)

Bases: object

The Bayesian quadrature method.

Bayesian quadrature solves integrals of the form

\[F = \int_\Omega f(x) d \mu(x).\]

Parameters:

• kernel (Kernel) – The kernel used for the Gaussian process model.

• measure (IntegrationMeasure) – The integration measure.

• policy (Optional[Policy]) – The policy choosing nodes at which to evaluate the integrand.

• belief_update (BQBeliefUpdate) – The inference method.

• stopping_criterion (BQStoppingCriterion) – The criterion that determines convergence.

• initial_design (Optional[InitialDesign]) – The initial design chooses a set of nodes once, before the acquisition loop with the policy runs.

Raises:

ValueError – If initial_design is given but policy is not given.

See also

bayesquad :

Computes the integral using an acquisition policy.

bayesquad_from_data :

Computes the integral \(F\) using a given dataset of nodes and function evaluations.

Methods Summary

bq_iterator(bq_state, info, fun, rng)	Generator that implements the iteration of the BQ method.
from_problem(input_dim[, kernel, measure, ...])	Creates an instance of this class from a problem description.
integrate(fun, nodes, fun_evals[, rng])	Integrates a given function.

Methods Documentation

bq_iterator(bq_state, info, fun, rng)

Generator that implements the iteration of the BQ method.

This function exposes the state of the BQ method one step at a time while running the loop.

Parameters:

• bq_state (BQState) – State of the Bayesian quadrature methods. Contains the information about the problem and the BQ belief.

• info (BQIterInfo) – The state of the iteration.

• fun (Callable | None) – Function to be integrated. It needs to accept a shape=(n_eval, input_dim) np.ndarray and return a shape=(n_eval,) np.ndarray.

• rng (Generator | None) – The random number generator used for random methods.

Yields:

• new_integral_belief – Updated belief about the integral.

• new_bq_state – The updated state of the Bayesian quadrature belief.

• new_info – The updated state of the iteration.

Return type:

Tuple[Normal, BQState, BQIterInfo]

classmethod from_problem(input_dim, kernel=None, measure=None, domain=None, policy='bmc', initial_design=None, options=None)

Creates an instance of this class from a problem description.

Parameters:

• input_dim (IntLike) – The input dimension.

• kernel (CovarianceFunction | None) – The kernel used for the GP model. Defaults to the ExpQuad kernel.

• measure (IntegrationMeasure | None) – The integration measure. Defaults to the Lebesgue measure on the domain.

• domain (DomainLike | None) – The integration bounds. Obsolete if measure is given.

• policy (str | None) – The policy choosing nodes at which to evaluate the integrand. Choose None if you want to integrate from a fixed dataset.

• initial_design (str | None) – The initial design chooses a set of nodes once, before the acquisition loop with the policy runs.

• options (dict | None) –

  A dictionary with the following optional solver settings

  scale_estimationOptional[str]

  Estimation method to use to compute the scale parameter. Defaults to ‘mle’.

  max_evalsOptional[IntLike]

  Maximum number of evaluations as stopping criterion.

  var_tolOptional[FloatLike]

  Variance tolerance as stopping criterion.

  rel_tolOptional[FloatLike]

  Relative tolerance as stopping criterion.

  jitterOptional[FloatLike]

  Non-negative jitter to numerically stabilise kernel matrix inversion. Defaults to 1e-8.

  batch_sizeOptional[IntLike]

  Batch size used in node acquisition. Defaults to 1.

  n_initial_design_nodesOptional[IntLike]

  The number of nodes created by the initial design. Defaults to input_dim * 5 if an initial design is given.

  n_candidatesOptional[IntLike]

  The number of candidate nodes used by the policies that maximize an acquisition function by drawing random candidates. Defaults to 1e2. Applicable to policies ‘us_rand’, ‘mi_rand’ and ‘ivr_rand’.

  n_restartsOptional[IntLike]

  The number of restarts that the acquisition optimizer performs in order to find the maximizer. Defaults to 10. Applicable to policies ‘us’, ‘mi’ and ‘ivr’.

Returns:

An instance of this class.

Return type:

BayesianQuadrature

Raises:

• NotImplementedError – If an unknown policy or an unknown initial_design is given.

• ValueError – If neither domain nor measure is given.

See also

bayesquad :

For details on options for policy and initial_design.

integrate(fun, nodes, fun_evals, rng=None)

Integrates a given function.

The function may be given as a function handle fun and/or numerically in terms of fun_evals at fixed nodes nodes.

If a policy is defined this method calls the generator bq_iterator until the first stopping criterion is met. The initial design is evaluated in a batch prior to running bq_iterator.

If no policy is defined this method immediately stops after processing the given nodes.

Parameters:

• fun (Callable | None) – Function to be integrated. It needs to accept a shape=(n_eval, input_dim) np.ndarray and return a shape=(n_eval,) np.ndarray.

• nodes (ndarray | None) – shape=(n_eval, input_dim) – Optional nodes at which function evaluations are available as fun_evals from start.

• fun_evals (ndarray | None) – shape=(n_eval,) – Optional function evaluations at nodes available from the start.

• rng (Generator | None) – The random number generator used for random methods.

Returns:

• integral_belief – Posterior belief about the integral.

• bq_state – Final state of the Bayesian quadrature method.

Raises:

• ValueError – If neither the integrand function fun nor integrand evaluations fun_evals are given.

• ValueError – If dimension of nodes or fun_evals is incorrect, or if their shapes do not match.

• ValueError – If rng is not given but policy or initial_design requires it.

• ValueError – If a policy is available but fun is not given.

• ValueError – If no policy is available and no nodes are given.

Warns:

UserWarning – When no policy is given and fun is ignored.

Return type:

Tuple[Normal, BQState, BQIterInfo]

Notes

The initial design is evaluated prior to running the bq_iterator and hence may not obey the stopping criterion. For example, if stopping is induced via a maximum number of evaluations (max_evals) smaller than the batch size of the initial design, the initial design will be evaluated nevertheless.