An Optimization Approach of Deriving Bounds between Entropy and Error from Joint Distribution: Case Study for Binary Classifications

In this work, we propose a new approach of deriving the bounds between entropy and error from a joint distribution through an optimization means. The specific case study is given on binary classifications. Two basic types of classification errors are investigated, namely, the Bayesian and non-Bayesian errors. The consideration of non-Bayesian errors is due to the facts that most classifiers result in non-Bayesian solutions. For both types of errors, we derive the closed-form relations between each bound and error components. When Fano’s lower bound in a diagram of “Error Probability vs. Conditional Entropy” is realized based on the approach, its interpretations are enlarged by including non-Bayesian errors and the two situations along with independent properties of the variables. A new upper bound for the Bayesian error is derived with respect to the minimum prior probability, which is generally tighter than Kovalevskij’s upper bound.