Asymptotically Optimal Change Detection for Unnormalized Pre- and Post-Change Distributions

TL;DR

This paper introduces the LPA-CUSUM algorithm for change detection using unnormalized distributions, achieving asymptotic optimality by unbiasedly estimating log-ratios of normalizing constants through thermodynamic integration.

Contribution

It proposes a novel LPA-CUSUM method that estimates CUSUM statistics without normalization, ensuring asymptotic optimality in challenging physics-based scenarios.

Findings

LPA-CUSUM provides unbiased log-ratio estimates.

The method achieves asymptotic optimality.

Numerical results confirm effectiveness.

Abstract

This paper addresses the problem of detecting changes when only unnormalized pre- and post-change distributions are accessible. This situation happens in many scenarios in physics such as in ferromagnetism, crystallography, magneto-hydrodynamics, and thermodynamics, where the energy models are difficult to normalize. Our approach is based on the estimation of the Cumulative Sum (CUSUM) statistics, which is known to produce optimal performance. We first present an intuitively appealing approximation method. Unfortunately, this produces a biased estimator of the CUSUM statistics and may cause performance degradation. We then propose the Log-Partition Approximation Cumulative Sum (LPA-CUSUM) algorithm based on thermodynamic integration (TI) in order to estimate the log-ratio of normalizing constants of pre- and post-change distributions. It is proved that this approach gives an unbiased estimate of the log-partition function and the CUSUM statistics, and leads to an asymptotically optimal performance. Moreover, we derive a relationship between the required sample size for thermodynamic integration and the desired detection delay performance, offering guidelines for practical parameter selection. Numerical studies are provided demonstrating the efficacy of our approach.