# On one-sample Bayesian tests for the mean

## TL;DR

This paper introduces a Bayesian method for one-sample z- and t-tests based on Kullback-Leibler divergence, providing evidence for the null hypothesis and avoiding classical paradoxes, with theoretical and practical validation.

## Contribution

It presents a novel Bayesian testing approach using KL divergence and relative belief ratios, offering evidence for the null and addressing limitations of existing methods.

## Key findings

- Method provides evidence in favor of the null hypothesis.
- Avoids classical paradoxes like Lindley's paradox.
- Demonstrated effectiveness through examples.

## Abstract

This paper deals with a new Bayesian approach to the standard one-sample $z$- and $t$- tests. More specifically, let $x_1,\ldots,x_n$ be an independent random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. The goal is to test the null hypothesis $\mathcal{H}_0: \mu=\mu_1$ against all possible alternatives. The approach is based on using the well-known formula of the Kullbak-Leibler divergence between two normal distributions (sampling and hypothesized distributions selected in an appropriate way). The change of the distance from a priori to a posteriori is compared through the relative belief ratio (a measure of evidence). Eliciting the prior, checking for prior-data conflict and bias are also considered. Many theoretical properties of the procedure have been developed. Besides it's simplicity, and unlike the classical approach, the new approach possesses attractive and distinctive features such as giving evidence in favor of the null hypothesis. It also avoids several undesirable paradoxes, such as Lindley's paradox that may be encountered by some existing Bayesian methods. The use of the approach has been illustrated through several examples.