An analytical approach to Bayesian evidence computation

TL;DR

This paper introduces exact and approximate analytical formulas for Bayesian evidence calculation in cosmology, enabling more efficient model comparison without extensive numerical methods.

Contribution

It provides new analytical expressions for Bayesian evidence with Gaussian and non-Gaussian likelihoods, simplifying model selection in cosmology.

Findings

Exact formulas for Gaussian likelihoods with arbitrary correlations.

Approximate formulas for non-Gaussian likelihoods including skewness and kurtosis.

Analytical results are less precise than numerical methods but still practically useful.

Abstract

The Bayesian evidence is a key tool in model selection, allowing a comparison of models with different numbers of parameters. Its use in analysis of cosmological models has been limited by difficulties in calculating it, with current numerical algorithms requiring supercomputers. In this paper we give exact formulae for the Bayesian evidence in the case of Gaussian likelihoods with arbitrary correlations and top-hat priors, and approximate formulae for the case of likelihood distributions with leading non-Gaussianities (skewness and kurtosis). We apply these formulae to cosmological models with and without isocurvature components, and compare with results we previously obtained using numerical thermodynamic integration. We find that the results are of lower precision than the thermodynamic integration, while still being good enough to be useful.