Extremal properties of likelihood-ratio quantizers

The paper concerns a situation in which there are M hypotheses H₁,. . ., H_M, and in which Y is a random variable taking values in a set Y´, with a different probability distribution under each hypothesis. A quantizer γ:Y´→{1,. . ., D} is applied to form a quantized random variable γ(Y). The extreme points of the set of possible probability distributions of γ(Y) are characterized as γ ranges over all quantizers. Optimality properties of likelihood-ratio quantizers are then established for a very broad class of quantization problems, including problems involving the maximization of a distance measure as discussed by S.M. Ali and S.D. Silvey(1966)