# BCMA-ES II: revisiting Bayesian CMA-ES

## TL;DR

This paper revisits Bayesian CMA-ES, clarifies the differences between normal and inverse Wishart priors, and introduces a mixture model to unify both approaches, supported by numerical experiments.

## Contribution

It provides theoretical insights into the covariance expectations of normal and inverse Wishart priors and proposes a generalized mixture model for Bayesian CMA-ES.

## Key findings

- Expected covariance is lower with normal Wishart prior due to convexity.
- The mixture model unifies normal and inverse Wishart priors.
- Numerical experiments compare the performance of both models and the generalized approach.

## Abstract

This paper revisits the Bayesian CMA-ES and provides updates for normal Wishart. It emphasizes the difference between a normal and normal inverse Wishart prior. After some computation, we prove that the only difference relies surprisingly in the expected covariance. We prove that the expected covariance should be lower in the normal Wishart prior model because of the convexity of the inverse. We present a mixture model that generalizes both normal Wishart and normal inverse Wishart model. We finally present various numerical experiments to compare both methods as well as the generalized method.