# IntegralVarianceReduction¶

The normalized reduction of the integral variance.

The acquisition function is

$\begin{split}a(x) &= \mathfrak{v}^{-1}(\mathfrak{v} - \mathfrak{v}(x))\\ &= \frac{(\int \bar{k}(x', x)p(x')\mathrm{d}x')^2}{\mathfrak{v} v(x)}\\ &= \rho^2(x)\end{split}$

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

The value $$a(x)$$ is equal to the squared correlation $$\rho^2(x)$$ between the hypothetical observation at $$x$$ and the integral value. 1

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

References

1

Gessner et al. Active Multi-Information Source Bayesian Quadrature, UAI, 2019

Attributes Summary

 has_gradients Whether the acquisition function exposes gradients.

Methods Summary

 __call__(x, bq_state) Evaluates the acquisition function and optionally its gradients.

Attributes Documentation

Methods Documentation

__call__(x, bq_state)[source]

Evaluates the acquisition function and optionally its gradients.

Parameters
• x (ndarray) – shape=(batch_size, input_dim) – The nodes where the acquisition function is being evaluated.

• bq_state (BQState) – State of the BQ belief.

Returns

• acquisition_valuesshape=(batch_size, ) – The acquisition values at nodes x.