  Chernoff coefficient is an upper bound of Bayes error probability in
classification problem. In this paper, we will develop sharp Chernoff type
bound on Bayes error probability. The new bound is not only an upper bound but
also a lower bound of Bayes error probability up to a constant in a
non-asymptotic setting. Moreover, we will apply this result to community
detection in stochastic block model. As a clustering problem, the optimal error
rate of community detection can be characterized by our Chernoff type bound.
This can be formalized by deriving a minimax error rate over certain class of
parameter space, then achieving such error rate by a feasible algorithm employ
multiple steps of EM type updates.
