An Information Theoretic Analysis of Sequential Decision-Making

We provide a novel analysis of Wald's sequential probability ratio test based on information theoretic measures. This test is optimal in the sense that it yields the minimum mean decision time. To analyze the decision-making process we consider information densities enabling to represent the stochastic information content of the observations yielding a stochastic termination time of the test. We study the consequences of the optimality of the test on the change of the information density in time. As a corollary, we find that the optimality of the test implies that the conditional probability to decide for hypothesis $\mathcal{H}_1$ (or the counter-hypothesis $\mathcal{H}_0$) given that the test terminates at time instant $k$ is independent of time. Moreover, we evaluate the evolution of the mutual information between the binary variable to be tested and the decision variable of the Wald test.