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

TL;DR

This work tackles quickest change detection when only unnormalized pre- and post-change distributions are accessible, a common scenario in high-dimensional physics models. It introduces LPA-CUSUM, a TI-based extension of CUSUM that unbiasedly estimates the log-ratio of partition functions $\log(Z_0/Z_1)$ via an oracle producing $Y$ with variance $\sigma^2$, enabling asymptotically optimal performance. The authors establish false-alarm and delay guarantees, derive guidelines linking TI sample size to detection delay through the parameter $\\gamma$, and provide a detailed TI-based estimation framework with variance bounds. Numerical experiments on MVN and Boltzmann distributions illustrate that, with sufficient TI samples, LPA-CUSUM approaches CUSUM performance and can outperform SCUSUM when normalization constants are intractable. Overall, the approach enables reliable, near-optimal change detection in unnormalized, high-dimensional energy-based models with practical parameter recommendations.

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.

Figures (1)

• Figure 1: Comparison of Empirical CADD vs. ARL for EXP (left) and MVN (right).

Theorems & Definitions (16)

• Lemma 1: chow1971great
• Lemma 2
• Lemma 3
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Remark 2
• Lemma 4
• Lemma 5