Approximating non-Gaussian Bayesian partitions with normalising flows: statistics, inference and application to cosmology

TL;DR

This paper introduces a method using normalising flows to simplify Bayesian inference for non-Gaussian distributions, improving the computation of partition functions and posterior distributions, with applications in cosmology.

Contribution

It demonstrates how normalising flows can approximate non-Gaussian Bayesian partitions and derive analytical expressions for posteriors beyond Gaussian assumptions.

Findings

Normalising flows effectively approximate non-Gaussian posteriors.

Enhanced computation of partition functions, cumulants, and entropy.

Successful application to cosmological parameters from supernova data.

Abstract

Subject of this paper is the simplification of Markov chain Monte Carlo sampling as used in Bayesian statistical inference by means of normalising flows, a machine learning method which is able to construct an invertible and differentiable transformation between Gaussian and non-Gaussian random distributions. We use normalising flows to compute Bayesian partition functions for non-Gaussian distributions and show how normalising flows can be employed in finding analytical expressions for posterior distributions beyond the Gaussian limit. Flows offer advantages for the numerical evaluation of the partition function itself, as well as for cumulants and for the information entropy. We demonstrate how normalising flows in conjunction with Bayes partitions can be used in inference problems in cosmology and apply them to the posterior distribution for the matter density $\Omega_m$ and a dark energy equation of state parameter $w_0$ on the basis of supernova data.