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Conditional Score Learning for Quickest Change Detection in Markov Transition Kernels

Wuxia Chen, Taposh Banerjee, Vahid Tarokh

TL;DR

The paper tackles quickest change detection in Markov processes with unknown transition kernels by learning the conditional score $\nabla_{\mathbf{y}} \log p(\mathbf{y}|\mathbf{x})$ from paired samples, enabling likelihood-free detection. It introduces a conditional Hyvärinen score framework and a score-based CUSUM (SCUSUM) procedure, including a truncated variant for stability and bounded increments. Theoretical results establish exponential lower bounds on mean time to false alarm and asymptotic upper bounds on detection delay under Doeblin-type uniform ergodicity, using Hoeffding-type concentration for dependent data. Empirically, the approach is validated on synthetic Gaussian-transition Markov processes and real CMU Motion Capture data, showing accurate conditional-score learning and timely change detection across multiple scenarios. Overall, the work extends score-based quickest change detection from i.i.d. settings to dependent, high-dimensional processes with practical training and rigorous performance guarantees.

Abstract

We address the problem of quickest change detection in Markov processes with unknown transition kernels. The key idea is to learn the conditional score $\nabla_{\mathbf{y}} \log p(\mathbf{y}|\mathbf{x})$ directly from sample pairs $( \mathbf{x},\mathbf{y})$, where both $\mathbf{x}$ and $\mathbf{y}$ are high-dimensional data generated by the same transition kernel. In this way, we avoid explicit likelihood evaluation and provide a practical way to learn the transition dynamics. Based on this estimation, we develop a score-based CUSUM procedure that uses conditional Hyvarinen score differences to detect changes in the kernel. To ensure bounded increments, we propose a truncated version of the statistic. With Hoeffding's inequality for uniformly ergodic Markov processes, we prove exponential lower bounds on the mean time to false alarm. We also prove asymptotic upper bounds on detection delay. These results give both theoretical guarantees and practical feasibility for score-based detection in high-dimensional Markov models.

Conditional Score Learning for Quickest Change Detection in Markov Transition Kernels

TL;DR

The paper tackles quickest change detection in Markov processes with unknown transition kernels by learning the conditional score from paired samples, enabling likelihood-free detection. It introduces a conditional Hyvärinen score framework and a score-based CUSUM (SCUSUM) procedure, including a truncated variant for stability and bounded increments. Theoretical results establish exponential lower bounds on mean time to false alarm and asymptotic upper bounds on detection delay under Doeblin-type uniform ergodicity, using Hoeffding-type concentration for dependent data. Empirically, the approach is validated on synthetic Gaussian-transition Markov processes and real CMU Motion Capture data, showing accurate conditional-score learning and timely change detection across multiple scenarios. Overall, the work extends score-based quickest change detection from i.i.d. settings to dependent, high-dimensional processes with practical training and rigorous performance guarantees.

Abstract

We address the problem of quickest change detection in Markov processes with unknown transition kernels. The key idea is to learn the conditional score directly from sample pairs , where both and are high-dimensional data generated by the same transition kernel. In this way, we avoid explicit likelihood evaluation and provide a practical way to learn the transition dynamics. Based on this estimation, we develop a score-based CUSUM procedure that uses conditional Hyvarinen score differences to detect changes in the kernel. To ensure bounded increments, we propose a truncated version of the statistic. With Hoeffding's inequality for uniformly ergodic Markov processes, we prove exponential lower bounds on the mean time to false alarm. We also prove asymptotic upper bounds on detection delay. These results give both theoretical guarantees and practical feasibility for score-based detection in high-dimensional Markov models.

Paper Structure

This paper contains 27 sections, 4 theorems, 63 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Conditional Score Definition and Properties
  3. Hyvärinen Score Difference Between Conditional Models
  4. Drift Characterization via Conditional Fisher Divergence
  5. Conditional Score Learning
  6. Approximating $\nabla_{\mathbf{y}} \log p(\mathbf{y} | \mathbf{x})$ via Score Matching
  7. Change Point Model Formulation
  8. Optimal Algorithm
  9. Score-Based Change Detection in Markov Processes
  10. Stopping Rule Based on Hyvärinen Score Differences
  11. Truncated Stopping Rule and Drift Preservation
  12. Analysis of False Alarms
  13. Hoeffding’s Inequality for Markov Chains
  14. Exponential Lower Bound on Time to False Alarm
  15. Analysis of Detection Delay
  16. ...and 12 more sections

Key Result

Lemma 4.2

Let $\{X_n\}_{n \in \mathbb{N}}$ be a Markov chain satisfying Doeblin’s condition. Define the second-order process $\{B_n\}_{n \in \mathbb{N}} \triangleq \{(X_{n-1}, X_n)\}_{n \in \mathbb{N}}$. Then, the process $\{B_n\}_{n \in \mathbb{N}}$ also satisfies Doeblin’s condition.

Figures (6)

  • Figure 1: Top: Hyvärinen score difference. Bottom: Corresponding SCUSUM statistic.
  • Figure 2: Log-scale average false alarm intervals and lower bound.
  • Figure 3: Detection delay and upper bound.
  • Figure 4: Marker setup: front and back view. The balls represent markers; the thick colored segments represent bones CMU_MoCap.
  • Figure 5: CMU motion capture CMU_MoCap sequences under three scenarios: running (top), basketball (middle), and jumping (bottom). Each row presents a sequence of four frames captured at random but increasing time indices.
  • ...and 1 more figures

Theorems & Definitions (9)

  • Remark
  • Lemma 4.2
  • Remark
  • Remark
  • Lemma 6.1: Hoeffding's Inequality for Uniformly Ergodic Chains, glynn2002hoeffdingxian2016online
  • Theorem 6.2: Exponential Lower Bound on Time to False Alarm
  • proof
  • Theorem 7.1: Detection Delay Upper Bound
  • proof