Asymptotically Optimal Sequential Testing with Markovian Data

2026-06-16 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Research Methodology & Innovation · Depth: Expert, extended

Summary

A new framework for asymptotically optimal sequential hypothesis testing with Markovian data is presented, generalizing prior work to composite null and alternative hypotheses. Published on February 19, 2026, this research establishes the first non-asymptotic, instance-dependent lower bound on the expected stopping time for any valid sequential test under the alternative. This bound, detailed in Theorem 3.3, incorporates the stationary distribution and transition structure of the unknown Markov chain, improving upon existing asymptotic or suboptimal bounds. The authors propose an optimal test (Algorithm 1) whose expected stopping time matches this lower bound asymptotically as the Type-I error probability (α) approaches zero. The framework's utility is demonstrated through applications such as sequential detection of model misspecification in Markov Chain Monte Carlo (MCMC) samplers and testing structural properties, like linearity of transition dynamics, in Markov Decision Processes (MDPs).

Key takeaway

For research scientists developing or validating sequential hypothesis tests for Markovian data, this framework offers a robust approach for composite null and alternative hypotheses. You should consider implementing the proposed martingale-based test, which is asymptotically optimal and provides strong Type-I error control. This allows for efficient detection of issues like MCMC misspecification or non-linear MDP dynamics, preventing prolonged computation on biased or incorrect models.

Key insights

Optimal sequential testing for Markovian data requires instance-dependent lower bounds and martingale-based procedures for composite hypotheses.

Principles

• Composite null hypotheses are fundamentally harder to test than singleton nulls.
• Stationary-weighted KL divergence quantifies instance-dependent testing hardness.
• Poisson equation solutions bound auxiliary terms in expected stopping time analysis.

Method

The proposed optimal test constructs an empirical transition kernel, computes a martingale-based statistic, and stops when it exceeds a state-visitation-count-dependent boundary.

In practice

• Detect MCMC sampler misspecification against a target stationary distribution.
• Verify linearity of transition dynamics in Markov Decision Processes.

Topics

• Sequential Hypothesis Testing
• Markov Chain Analysis
• Asymptotic Optimality
• MCMC Misspecification Detection
• Markov Decision Processes
• Kullback-Leibler Divergence
• Poisson Equation

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.