# Comparison Between Bayesian and Frequentist Tail Probability Estimates

## TL;DR

This paper compares Bayesian and frequentist methods for estimating tail probabilities, revealing why Bayesian estimates tend to be higher, and establishes conditions based on distribution convexity using mathematical inequalities.

## Contribution

It provides a theoretical analysis explaining the consistent overestimation of tail probabilities by Bayesian methods compared to frequentist ones.

## Key findings

- Bayesian tail probability estimates are always higher than frequentist estimates under certain conditions.
- Convexity of the distribution function is key to understanding the difference.
- Mathematical inequalities like Jensen's are used to establish the conditions.

## Abstract

In this paper, we investigate the reasons that the Bayesian estimator of the tail probability is always higher than the frequentist estimator. Sufficient conditions for this phenomenon are established both by using Jensen's Inequality and by looking at Taylor series approximations, both of which point to the convexity of the distribution function.