"""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]`.
References
----------
.. [1] Gessner et al. Active Multi-Information Source Bayesian Quadrature,
*UAI*, 2019
"""
@property
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 -= np.dot(bq_state.kernel_means, weights)
values = (bq_state.scale_sq * predictive_embedding) ** 2 / (
bq_state.integral_belief.cov * y_predictive_var
)
return values, None