  We consider the problem of estimating the probability of error in
multi-hypothesis testing when MAP criterion is used. This probability, which is
also known as the Bayes risk is an important measure in many communication and
information theory problems. In general, the exact Bayes risk can be difficult
to obtain. Many upper and lower bounds are known in literature. One such upper
bound is the equivocation bound due to R\'enyi which is of great philosophical
interest because it connects the Bayes risk to conditional entropy. Here we
give a simple derivation for an improved equivocation bound.
  We then give some typical examples of problems where these bounds can be of
use. We first consider a binary hypothesis testing problem for which the exact
Bayes risk is difficult to derive. In such problems bounds are of interest.
Furthermore using the bounds on Bayes risk derived in the paper and a random
coding argument, we prove a lower bound on equivocation valid for most random
codes over memoryless channels.
