Information–Theoretic Applications of the Logarithmic Probability Comparison Bound

A well-known technique in estimating the probabilities of rare events in general and in information theory in particular (used, for example, in the sphere–packing bound) is that of finding a reference probability measure under which the event of interest has the probability of order one and estimating the probability in question by means of the Kullback–Leibler divergence. A method has recently been proposed in [2] that can be viewed as an extension of this idea in which the probability under the reference measure may itself be decaying exponentially, and the Rényi divergence is used instead. The purpose of this paper is to demonstrate the usefulness of this approach in various information–theoretic settings. For the problem of channel coding, we provide a general methodology for obtaining matched, mismatched, and robust error exponent bounds, as well as new results in a variety of particular channel models. Other applications we address include rate-distortion coding and the problem of guessing.