Evaluation of Error Probability of Classification Based on the Analysis of the Bayes Code

Abstract

Suppose that we have two training sequences generated by parametrized distributions ${P_{\theta _1^{\ast}}}$ and ${P_{\theta _2^{\ast}}}$, where $\theta _1^{\ast}$ and $\theta _2^{\ast}$ are unknown. Given training sequences, we study the problem of classifying whether a test sequence was generated according to ${P_{\theta _1^{\ast}}}$ or ${P_{\theta _2^{\ast}}}$. This problem can be thought of as a hypothesis testing problem and the weighted sum of type-I and type-II error probabilities is analyzed. To prove the results, we utilize the analysis of the codeword lengths of the Bayes code. It is shown that upper and lower bounds of the probability of error are characterized by the terms containing the Chernoff information, the dimension of a parameter space, and the ratio of the length between the training sequences and the test sequence. Further, we generalize the part of the preceding results to multiple hypotheses setup.