A Linear Programming Approach to Sequential Hypothesis Testing

TL;DR

This paper introduces a linear programming framework for designing optimal sequential hypothesis tests under Markov assumptions, enabling efficient computation of tests for Gaussian processes with various dependencies.

Contribution

It formulates the sequential testing problem as a linear program using Lagrangian duality, providing a novel computational approach.

Findings

Linear programming approach successfully applied to Gaussian processes

Optimal tests derived for Gaussian AR(1) and other dependency structures

Method enables efficient computation of sequential tests

Abstract

Under some mild Markov assumptions it is shown that the problem of designing optimal sequential tests for two simple hypotheses can be formulated as a linear program. The result is derived by investigating the Lagrangian dual of the sequential testing problem, which is an unconstrained optimal stopping problem, depending on two unknown Lagrangian multipliers. It is shown that the derivative of the optimal cost function with respect to these multipliers coincides with the error probabilities of the corresponding sequential test. This property is used to formulate an optimization problem that is jointly linear in the cost function and the Lagrangian multipliers and an be solved for both with off-the-shelf algorithms. To illustrate the procedure, optimal sequential tests for Gaussian random sequences with different dependency structures are derived, including the Gaussian AR(1) process.