مرکز منطقه ای اطلاع رساني علوم و فناوري - On the relationship between the separability measures and the Bayes probability of error

Abstract :

The information measures, as a special class of efficiency measures of muiticategory information systems, and their relations to the Bayes probability of error $P_{B}$ have been recently defined and investigated. Another class of efficiency measures called the separability measures is introduced in this paper. The relationship between any separability measure and $P_{B}$ is determined for any $2 \\leq Q < \\infty$ , where $Q$ denotes the number of categories in a multicategory information system. As before, $\\epsilon_{0}$ and $\\epsilon_{m}$ criteria are proposed as the similarity measures between the separability measures and $P_{B}$ . The problems of determination, for any $2 \\leq Q < \\infty$ , of all the separability measures with minimal $\\epsilon_{0}$ and $\\epsilon_{m}$ criteria, called $\\epsilon_{0}$ -optimal and $\\epsilon_{m}$ -Optimal, are defined and completely solved, respectively. The minimal values of $\\epsilon_{0}$ and $\\epsilon_{m}$ criteria are evaluated as well. It is proved that the information measures are for each $3 \\leq Q < \\infty$ more similar to $P_{B}$ than the separability measures with respect to the minimal values of both $\\epsilon_{0}$ and $\\epsilon_{m}$ criteria. It is pointed out that the average conditional quadratic entropy is not only an information measure but also a separability measure, which is, for each $3\\leq Q < \\infty , \\epsilon_{0}$ -optimal and very close to $\\epsilon_{m}$ -optimal separability measures with respect to the $\\epsilon_{m}$ criterion.