On the relationship between the information measures and the Bayes probability of error

The concept of efficiency measures of multicategory information systems as well as the concepts of the relationship and the similarity measures between two efficiency measures have been recently introduced and developed. In this paper, the concave measures as a general class of efficiency measures and the information measures as a special class of concave measures are defined and investigated. The relationship between any concave measure, information measure in particular, and the Bayes probability of error
P_{B}
is determined for any
2\leq Q < \infty
, where
Q
denotes the number of categories in a multicategory information system. The so-called
\epsilon_{ }
0 and
\epsilon_{m}
criteria are proposed as the similarity measures between the information measures and
P_{B}
. The problems of determination, for any
2 \leq Q < \infty
, of all the information measures with minimal
\epsilon_{0}
and
\epsilon_{m}
criteria, called
\epsilon_{0}
-optimal and
\epsilon_{m}
-optimal, are formulated and completely solved, respectively. Additionally, the minimal values of
\epsilon_{0}
and
\epsilon_{m}
criteria are evaluated as well. It is pointed out that the well-known average conditional quadratic entropy is for all
2 \leq Q < \infty
very close to the
\epsilon_{0}
-optimal and
\epsilon_{m}
-Optimal information measures with respect to
\epsilon_{0}
and
\epsilon_{m}
criteria, respectively.