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The existing upper and lower bounds between entropy and error probability aremostly derived from the inequality of the entropy relations, which couldintroduce approximations into the analysis. We derive analytical bounds basedon the closed-form solutions of conditional entropy without involving anyapproximation. Two basic types of classification errors are investigated in thecontext of binary classification problems, namely, Bayesian and non-Bayesianerrors. We theoretically confirm that Fano's lower bound is an exact lowerbound for any types of classifier in a relation diagram of "error probabilityvs. conditional entropy". The analytical upper bounds are achieved with respectto the minimum prior probability, which are tighter than Kovalevskij's upperbound.

Analytical Bounds between Entropy and Error Probability in Binary Classifications