Bayesian sequential composite hypothesis testing in discrete time

TL;DR

This paper investigates Bayesian sequential hypothesis testing for an unknown parameter in exponential families, demonstrating the concavity of the cost function and monotonicity of optimal stopping boundaries over time.

Contribution

It introduces a Markovian framework for Bayesian sequential testing, proving cost function concavity and monotonicity of stopping boundaries for various models.

Findings

Cost function is concave in the Bayesian setting.

Posterior distribution becomes more concentrated over time.

Optimal stopping boundaries are monotone in many models.

Abstract

We study the sequential testing problem of two alternative hypotheses regarding an unknown parameter in an exponential family when observations are costly. In a Bayesian setting, the problem can be embedded in a Markovian framework. Using the conditional probability of one of the hypotheses as the underlying spatial variable, we show that the cost function is concave and that the posterior distribution becomes more concentrated as time goes on. Moreover, we study time monotonicity of the value function. For a large class of model specifications, the cost function is non-decreasing in time, and the optimal stopping boundaries are thus monotone.