A uniÞed framework for evaluating bounds on the Bayesian cost

Abstract

A unified framework for evaluating bounds on the average cost of an optimum Bayesian receiver with arbitrary cost assignments is presented. The framework is developed based on formulating the binary hypothesis testing problem from a decision-theoretic perspective. This formulation results in a representation for the minimum average cost that is analogous to that for the minimum probability of error. Taking advantage of this analogy, a whole new series of generalized bounds on the minimum average cost is obtained by employing the well-developed theory of bounds of the minimum probability of error problem available in the literature. To demonstrate the applicability of the proposed unified framework, two upper bounds on the minimum cost, that generalize the known Bhattacharyya and Chernoff upper bounds on the minimum probability of error, are derived. The unified framework is also used to obtain a new generalized class of upper and lower bounds in terms of a modified form of the f-divergence. All new bounds derived in the paper are shown to reduce to the probability of error bounds under special cost assignments.