On a Relationship between Bayesian Decision and Estimation

This paper presents several new results that give more insight on a general relationship between optimal Bayesian decision and estimation. For a standard signal classification problem the maximum a posteriori probability (MAP) decision is decomposed as a cascade of the continuous-valued minimum mean square error (MMSE) estimator (of an appropriately defined vector indicator) and a vector quantization operator for the obtained estimate. An explicit geometric characterization of this vector quantization operator is obtained. Furthermore, it is shown that the same quantization operator can be also implemented through nearest-neighbor quantization with an appropriately chosen vector norm.