# Arimoto-R\'enyi Conditional Entropy and Bayesian $M$-ary Hypothesis Testing

## TL;DR

This paper explores bounds on the minimum error probability in Bayesian M-ary hypothesis testing using Arimoto-Rényi conditional entropy, extending classical inequalities and analyzing their implications for channel coding and error decay.

## Contribution

It introduces new bounds on error probability based on Arimoto-Rényi entropy, generalizes Fano's inequality, and analyzes error decay in discrete memoryless channels.

## Key findings

- Tighter bounds on error probability using Arimoto-Rényi entropy.
- Generalization of Fano's inequality for finite and infinite hypotheses.
- Analysis of error decay in channel coding scenarios.

## Abstract

This paper gives upper and lower bounds on the minimum error probability of Bayesian $M$-ary hypothesis testing in terms of the Arimoto-R\'enyi conditional entropy of an arbitrary order $\alpha$. The improved tightness of these bounds over their specialized versions with the Shannon conditional entropy ($\alpha=1$) is demonstrated. In particular, in the case where $M$ is finite, we show how to generalize Fano's inequality under both the conventional and list-decision settings. As a counterpart to the generalized Fano's inequality, allowing $M$ to be infinite, a lower bound on the Arimoto-R\'enyi conditional entropy is derived as a function of the minimum error probability. Explicit upper and lower bounds on the minimum error probability are obtained as a function of the Arimoto-R\'enyi conditional entropy for both positive and negative $\alpha$. Furthermore, we give upper bounds on the minimum error probability as functions of the R\'enyi divergence. In the setup of discrete memoryless channels, we analyze the exponentially vanishing decay of the Arimoto-R\'enyi conditional entropy of the transmitted codeword given the channel output when averaged over a random coding ensemble.