probnum.quad.bayesquad(fun, input_dim, kernel=None, domain=None, measure=None, policy='bmc', max_evals=None, var_tol=None, rel_tol=None, batch_size=1, rng=Generator(PCG64) at 0x7F92DAFAC200)[source]

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 (int) – Input dimension of the integration problem.

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

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

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

• policy (Optional[str]) –

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

 Bayesian Monte Carlo 2 bmc

• max_evals (Optional[IntLike]) – Maximum number of function evaluations.

• var_tol (Optional[FloatLike]) – Tolerance on the variance of the integral.

• rel_tol (Optional[FloatLike]) – Tolerance on consecutive updates of the integral mean.

• batch_size (Optional[IntLike]) – Number of new observations at each update.

• rng (Optional[np.random.Generator]) – Random number generator. Used by Bayesian Monte Carlo other random sampling policies. Optional. Default is np.random.default_rng().

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

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.

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.

Examples

>>> import numpy as np

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