Evaluation of Error Probability of Classification Based on the Analysis of the Bayes Code: Extension and Example

Suppose that we have two training sequences generated by parametrized distributions $P_{\theta^*}$ and $P_{\xi^*}$, where $\theta^*$ and $\xi^*$ are unknown true parameters. Given training sequences, we study the problem of classifying whether a test sequence was generated according to $P_{\theta^*}$ or $P_{\xi^*}$. This problem can be thought of as a hypothesis testing problem and our aim is to analyze the weighted sum of type-I and type-II error probabilities. Utilizing the analysis of the codeword lengths of the Bayes code, our previous study derived more refined bounds on the error probability than known previously. However, our previous study had the following deficiencies: i) the prior distributions of $\theta$ and $\xi$ are the same; ii) the prior distributions of two hypotheses are uniform; iii) no numerical calculation at finite blocklength. This study solves these problems. We remove the restrictions i) and ii) and derive more general results than obtained previously. To deal with problem iii), we perform a numerical calculation for a concrete model.