A Linear Programming Approach to Sequential Hypothesis Testing

• Published in: Sequential analysis Vol. 34; no. 2; pp. 235 - 263
• Main Authors: Fauß, Michael, Zoubir, Abdelhak M.
• Format: Journal Article
• Language: English
• Published: Philadelphia Taylor & Francis Group 03.04.2015; Taylor & Francis Ltd
• Subjects: Algorithms; Cost function; Derivatives; Discrete-time Markov process; Gaussian; Hypothesis testing; Linear programming; Markov analysis; Markov processes; Multipliers; Normal distribution; Optimal sequential test; Optimal stopping; Optimization; Sequential hypothesis testing; Two simple hypotheses
• ISSN: 0747-4946; 1532-4176
• DOI: 10.1080/07474946.2015.1030981

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. This 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 can 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.