bayesquad

probnum.quad.bayesquad(fun, input_dim, kernel=None, measure=None, domain=None, policy='bmc', initial_design=None, rng=None, options=None)

Infer the solution of the uni- or multivariate integral \(\int_\Omega f(x) d \mu(x)\) on a hyper-rectangle \(\Omega = [a_1, b_1] \times \cdots \times [a_D, b_D]\) or \(\Omega = \mathbb{R}^D\).

Bayesian quadrature (BQ) infers integrals of the form

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

of a function \(f:\mathbb{R}^D \mapsto \mathbb{R}\) integrated on the domain \(\Omega \subset \mathbb{R}^D\) against a measure \(\mu\) on \(\mathbb{R}^D\).

Bayesian quadrature methods return a probability distribution over the solution \(F\) with uncertainty arising from finite computation (here a finite number of function evaluations). They start out with a random process encoding the prior belief about the function \(f\) to be integrated. Conditioned on either existing or acquired function evaluations according to a policy, they update the belief on \(f\), which is translated into a posterior measure over the integral \(F\). See Briol et al. 1 for a review on Bayesian quadrature.

Parameters

  • fun (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.

  • input_dim (IntLike) – Input dimension of the integration problem.

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

  • measure (Optional[IntegrationMeasure]) – The integration measure. Defaults to the Lebesgue measure on domain.

  • domain (Optional[DomainLike]) – The integration domain. Contains lower and upper bound as scalar or np.ndarray. Obsolete if measure is given.

  • policy (Optional[str]) –

    Type of acquisition strategy to use. Defaults to ‘bmc’. Options are

    Bayesian Monte Carlo 2

    bmc

    Van Der Corput points

    vdc

    Uncertainty Sampling with random candidates

    us_rand

    Uncertainty Sampling with optimizer

    us

  • initial_design (Optional[str]) –

    The type of initial design to use. If None is given, no initial design is used. Options are

    Samples from measure

    mc

    Latin hypercube 3

    latin

  • rng (Optional[np.random.Generator]) – The random number generator used for random methods.

  • options (Optional[dict]) –

    A dictionary with the following optional solver settings

    scale_estimationOptional[str]

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

    Maximum likelihood estimation

    mle
    max_evalsOptional[IntLike]

    Maximum number of function evaluations.

    var_tolOptional[FloatLike]

    Tolerance on the variance of the integral.

    rel_tolOptional[FloatLike]

    Tolerance on consecutive updates of the integral mean.

    jitterOptional[FloatLike]

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

    batch_sizeOptional[IntLike]

    Number of new observations at each update. 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.

    us_rand_n_candidatesOptional[IntLike]

    The number of candidate nodes used by the policy ‘us_rand’. Defaults to 1e2.

    us_n_restartsOptional[IntLike]

    The number of restarts that the acquisition optimizer performs in order to find the maximizer when policy ‘us’ is used. Defaults to 10.

Returns

  • integral – The integral belief of \(F\) subject to the provided measure or domain.

  • info – Information on the performance of the method.

Raises

ValueError – If neither a domain nor a measure are given.

Warns

UserWarning – When domain is given but not used.

Return type

Tuple[Normal, BQIterInfo]

Notes

If multiple stopping conditions are provided, the method stops once one of them is satisfied. If no stopping condition is provided, the default values are max_evals = 25 * input_dim and var_tol = 1e-6.

See also

bayesquad_from_data

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

References

1

Briol, F.-X., et al., Probabilistic integration: A role in statistical computation?, Statistical Science 34.1, 2019, 1-22, 2019

2

Rasmussen, C. E., and Z. Ghahramani, Bayesian Monte Carlo, Advances in Neural Information Processing Systems, 2003, 505-512.

3

Mckay et al., A Comparison of Three Methods for Selecting Values of Input Variables in the Analysis of Output from a Computer Code, Technometrics, 1979.

Examples

>>> import numpy as np

>>> input_dim = 1
>>> domain = (0, 1)
>>> def fun(x):
...     return x.reshape(-1, )
>>> F, info = bayesquad(fun, input_dim, domain=domain, rng=np.random.default_rng(0))
>>> print(F.mean)
0.5