[Paper Review] Minimax Optimal Sequential Hypothesis Tests for Markov Processes

Michael Fauß, Abdelhak M. Zoubir+1 more|arXiv (Cornell University)|Nov 10, 2018

Advanced Statistical Process MonitoringDecision Sciences58 references3 citations

TL;DR

This paper develops minimax optimal sequential hypothesis tests for Markov processes under distributional uncertainty, combining sequential testing efficiency with robustness to model mismatch. It establishes sufficient conditions for strict minimax optimality by identifying least favorable distributions as those maximally similar under a data-dependent generalized f-dissimilarity, enabling robust, efficient decision-making in time-critical applications with uncertain models.

ABSTRACT

Under mild Markov assumptions, sufficient conditions for strict minimax optimality of sequential tests for multiple hypotheses under distributional uncertainty are derived. First, the design of optimal sequential tests for simple hypotheses is revisited and it is shown that the partial derivatives of the corresponding cost function are closely related to the performance metrics of the underlying sequential test. Second, an implicit characterization of the least favorable distributions for a given testing policy is stated. By combining the results on optimal sequential tests and least favorable distributions, sufficient conditions for a sequential test to be minimax optimal under general distributional uncertainties are obtained. The cost function of the minimax optimal test is further identified as a generalized $f$-dissimilarity and the least favorable distributions as those that are most similar with respect to this dissimilarity. Numerical examples for minimax optimal sequential tests under different uncertainties illustrate the theoretical results.

Motivation & Objective

• To develop minimax optimal sequential tests for multiple hypotheses in Markov processes under distributional uncertainty.
• To address the limitation of classical sequential tests that degrade under model mismatch by incorporating robustness into the design.
• To unify the Kiefer–Weiss problem and robust detection within a single framework for sequential testing.
• To characterize least favorable distributions as those maximally similar under a generalized f-dissimilarity with data-dependent weights.
• To provide a practical design method for minimax optimal tests with numerical validation.

Proposed method

• Derives sufficient conditions for strict minimax optimality under mild Markov assumptions and general distributional uncertainty.
• Revisits optimal sequential tests for multiple simple hypotheses, linking cost function partial derivatives to performance metrics.
• Characterizes least favorable distributions implicitly, showing they arise when the underlying process becomes Markovian under data-dependent weighting.
• Identifies the cost function of the minimax optimal test as a generalized f-dissimilarity, with least favorable distributions being those most similar under this measure.
• Proposes a practical design method using duality and saddle-point conditions, validated via numerical examples.
• Establishes equivalence between minimax optimality and saddle-point solutions involving dual variables and constrained error probabilities.

Experimental results

Research questions

• RQ1 Under what conditions is a sequential test minimax optimal for multiple hypotheses under distributional uncertainty in Markov processes?
• RQ2 How can the least favorable distributions be characterized in the context of sequential hypothesis testing with model uncertainty?
• RQ3 What statistical similarity measure governs the identification of least favorable distributions in the minimax framework?
• RQ4 How do data-dependent weights influence the robustness and optimality of sequential tests?
• RQ5 Can the minimax optimal test be designed practically, and how does it compare to classical approaches under model mismatch?

Key findings

• Sufficient conditions for strict minimax optimality are derived for sequential tests under mild Markov assumptions and general distributional uncertainty.
• The minimax optimal test is shown to be optimal for least favorable distributions, which are implicitly characterized via the test's performance metrics.
• Least favorable distributions are identified as those maximally similar under a generalized f-dissimilarity, with weights that depend on the data and test policy.
• The cost function of the minimax optimal test is formally identified as a generalized f-dissimilarity, providing a statistical foundation for robustness.
• Numerical examples demonstrate the effectiveness of the proposed method in maintaining low expected run-length and error probabilities under model mismatch.
• The framework unifies the Kiefer–Weiss problem and robust detection, showing that minimax optimal sequential tests can be derived via duality and saddle-point analysis.