# Robust Mean Estimation with the Bayesian Median of Means

## TL;DR

This paper proposes the Bayesian median of means, a new aggregation method that reduces variance in estimates at the cost of bias, demonstrating improved performance in high-variance scenarios through theoretical analysis and empirical validation.

## Contribution

Introduction of the Bayesian median of means, a non-parametric aggregation rule that interpolates between mean and median, with theoretical guarantees and practical algorithms.

## Key findings

- Significantly reduces variance in high-variance settings.
- Maintains consistency and asymptotically negligible bias.
- Performs well in real-world applications like importance sampling and cross-validation.

## Abstract

The sample mean is often used to aggregate different unbiased estimates of a parameter, producing a final estimate that is unbiased but possibly high-variance. This paper introduces the Bayesian median of means, an aggregation rule that roughly interpolates between the sample mean and median, resulting in estimates with much smaller variance at the expense of bias. While the procedure is non-parametric, its squared bias is asymptotically negligible relative to the variance, similar to maximum likelihood estimators. The Bayesian median of means is consistent, and concentration bounds for the estimator's bias and $L_1$ error are derived, as well as a fast non-randomized approximating algorithm. The performances of both the exact and the approximate procedures match that of the sample mean in low-variance settings, and exhibit much better results in high-variance scenarios. The empirical performances are examined in real and simulated data, and in applications such as importance sampling, cross-validation and bagging.