Source code for probnum.quad.solvers.acquisition_functions._integral_variance_reduction

"""Integral variance reduction acquisition function for Bayesian quadrature."""

from __future__ import annotations

from typing import Optional, Tuple

import numpy as np

from probnum.quad.solvers._bq_state import BQState
from probnum.quad.solvers.belief_updates import BQStandardBeliefUpdate

from ._acquisition_function import AcquisitionFunction

# pylint: disable=too-few-public-methods

class IntegralVarianceReduction(AcquisitionFunction):
    r"""The normalized reduction of the integral variance.

    The acquisition function is

    .. math::
        a(x) &= \mathfrak{v}^{-1}(\mathfrak{v} - \mathfrak{v}(x))\\
             &= \frac{(\int \bar{k}(x', x)p(x')\mathrm{d}x')^2}{\mathfrak{v} v(x)}\\
             &= \rho^2(x)

    where :math:`\mathfrak{v}` is the current integral variance, :math:`\mathfrak{v}(x)`
    is the integral variance including a hypothetical observation at
    :math:`x`, :math:`v(x)` is the predictive variance for :math:`f(x)` and
    :math:`\bar{k}(x', x)` is the posterior kernel function.

    The value :math:`a(x)` is equal to the squared correlation :math:`\rho^2(x)` between
    the hypothetical observation at :math:`x` and the integral value. [1]_

    The normalization constant :math:`\mathfrak{v}^{-1}` ensures that
    :math:`a(x)\in[0, 1]`.

    .. [1] Gessner et al. Active Multi-Information Source Bayesian Quadrature,
       *UAI*, 2019


    def has_gradients(self) -> bool:
        # Todo (#581): this needs to return True, once gradients are available
        return False

[docs] def __call__( self, x: np.ndarray, bq_state: BQState, ) -> Tuple[np.ndarray, Optional[np.ndarray]]: _, y_predictive_var = BQStandardBeliefUpdate.predict_integrand(x, bq_state) # if observation noise is added to BQ, it needs to be retrieved here. observation_noise_var = 0.0 # dummy placeholder y_predictive_var += observation_noise_var predictive_embedding = bq_state.kernel_embedding.kernel_mean(x) # posterior if observations are available if bq_state.fun_evals.shape[0] > 0: weights = BQStandardBeliefUpdate.gram_cho_solve( bq_state.gram_cho_factor, bq_state.kernel.matrix(bq_state.nodes, x) ) predictive_embedding -=, weights) values = (bq_state.scale_sq * predictive_embedding) ** 2 / ( bq_state.integral_belief.cov * y_predictive_var ) return values, None