Approximation of differential entropy in Bayesian optimal experimental design
This work addresses computational bottlenecks in large-scale inference problems like inverse problems, where expensive likelihood evaluations dominate costs, offering an incremental improvement over existing methods.
The paper tackles the computational challenge of estimating expected information gain in Bayesian optimal experimental design by reducing it to maximum entropy estimation when differential entropy is independent of design or explicitly evaluable, achieving convergence rates comparable to or better than state-of-the-art methods with negligible entropy evaluation cost.
Bayesian optimal experimental design provides a principled framework for selecting experimental settings that maximize obtained information. In this work, we focus on estimating the expected information gain in the setting where the differential entropy of the likelihood is either independent of the design or can be evaluated explicitly. This reduces the problem to maximum entropy estimation, alleviating several challenges inherent in expected information gain computation. Our study is motivated by large-scale inference problems, such as inverse problems, where the computational cost is dominated by expensive likelihood evaluations. We propose a computational approach in which the evidence density is approximated by a Monte Carlo or quasi-Monte Carlo surrogate, while the differential entropy is evaluated using standard methods without additional likelihood evaluations. We prove that this strategy achieves convergence rates that are comparable to, or better than, state-of-the-art methods for full expected information gain estimation, particularly when the cost of entropy evaluation is negligible. Moreover, our approach relies only on mild smoothness of the forward map and avoids stronger technical assumptions required in earlier work. We also present numerical experiments, which confirm our theoretical findings.