BayesianQuadrature

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

Bases: object

A base class for Bayesian quadrature.

Bayesian quadrature solves integrals of the form

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

Parameters

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

• measure (IntegrationMeasure) – The integration measure.

• policy (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.

Methods Summary

bq_iterator([fun, nodes, fun_evals, ...])	Generator that implements the iteration of the BQ method.
from_problem(input_dim[, kernel, measure, ...])	Alternative way to initialize Bayesian_Quadrature
has_converged(bq_state)	Checks if the BQ method has converged.
integrate([fun, nodes, fun_evals])	Integrate the function fun.

Methods Documentation

bq_iterator(fun=None, nodes=None, fun_evals=None, integral_belief=None, bq_state=None)

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

• fun (Optional[Callable]) – 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 (Optional[ndarray]) – shape=(n_eval, input_dim) – Optional nodes at which function evaluations are available as fun_evals from start.

• fun_evals (Optional[ndarray]) – shape=(n_eval,) – Optional function evaluations at nodes available from the start.

• integral_belief (Optional[Normal]) – Current belief about the integral.

• bq_state (Optional[BQState]) – State of the Bayesian quadrature methods. Contains all necessary information about the problem and the computation.

Yields

• new_integral_belief – Updated belief about the integral.

• new_nodes – shape=(n_new_eval, input_dim) – The new location(s) at which new_fun_evals are available found during the iteration.

• new_fun_evals – shape=(n_new_eval,) – The function evaluations at the new locations new_nodes.

• new_bq_state – Updated state of the Bayesian quadrature methods.

Return type

Tuple[Normal, ndarray, ndarray, BQState]

classmethod from_problem(input_dim, kernel=None, measure=None, domain=None, policy='bmc', max_evals=None, var_tol=None, rel_tol=None, batch_size=1, rng=None)

Alternative way to initialize Bayesian_Quadrature

Parameters

• input_dim (int) – Input dimension.

• kernel (Optional[Kernel]) – The kernel used for the GP model.

• measure (Optional[IntegrationMeasure]) – The integration measure.

• domain (Union[Tuple[Union[float, Real, floating], Union[float, Real, floating]], Tuple[ndarray, ndarray], None]) – The integration bounds.

• policy (str) – The policy choosing nodes at which to evaluate the integrand.

• max_evals (Union[int, Integral, integer, None]) – Maximum number of evaluations as stopping criterion.

• var_tol (Union[float, Real, floating, None]) – Variance tolerance as stopping criterion.

• rel_tol (Union[float, Real, floating, None]) – Relative tolerance as stopping criterion.

• batch_size (Union[int, Integral, integer]) – Batch size used in node acquisition.

• rng (Optional[Generator]) – The random number generator.

Returns

An instance of this class.

Return type

BayesianQuadrature

Raises

• ValueError – If Bayesian Monte Carlo (‘bmc’) is selected as policy and no random number generator (rng) is given.

• NotImplementedError – If an unknown policy is given.

has_converged(bq_state)

Checks if the BQ method has converged.

Parameters

bq_state (BQState) – State of the Bayesian quadrature methods. Contains all necessary information about the problem and the computation.

Returns

Whether the solver has converged.

Return type

has_converged

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

Integrate the function fun.

fun may be analytically given, or numerically in terms of fun_evals at fixed nodes. This function calls the generator bq_iterator until the first stopping criterion is met.

Parameters

• fun (Optional[Callable]) – 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 (Optional[ndarray]) – shape=(n_eval, input_dim) – Optional nodes at which function evaluations are available as fun_evals from start.

• fun_evals (Optional[ndarray]) – shape=(n_eval,) – Optional function evaluations at nodes available from the start.

Return type

Tuple[Normal, BQState]

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.