Partition function approach to non-Gaussian likelihoods: Formalism and expansions for weakly non-Gaussian cosmological inference

TL;DR

This paper introduces a partition function framework for analyzing non-Gaussian likelihoods in cosmology, providing a formalism that simplifies the derivation of cumulants and connects to existing expansion methods, with practical demonstrations.

Contribution

It develops a novel partition function approach for weakly non-Gaussian likelihoods, linking it to Fisher, DALI, and Gram-Charlier expansions, and explores extensions to Hamilton Monte-Carlo and ensemble methods.

Findings

Validates the approach with a supernova-likelihood example

Shows the connection to Fisher matrix and higher-order cumulants

Discusses extensions to Hamilton Monte-Carlo and ensemble methods

Abstract

Non-Gaussian likelihoods, ubiquitous throughout cosmology, are a direct consequence of nonlinearities in the physical model. Their treatment requires Monte-Carlo Markov-chain or more advanced sampling methods for the determination of confidence contours. As an alternative, we construct canonical partition functions as Laplace-transforms of the Bayesian evidence, from which MCMC-methods would sample microstates. Cumulants of order $n$ of the posterior distribution follow by direct $n$-fold differentiation of the logarithmic partition function, recovering the classic Fisher-matrix formalism at second order. We connect this approach for weakly non-Gaussianities to the DALI- and Gram-Charlier expansions and demonstrate the validity with a supernova-likelihood on the cosmological parameters $\Omega_m$ and $w$. We comment on extensions of the canonical partition function to include kinetic energies in order to bridge to Hamilton Monte-Carlo sampling, and on ensemble Markov-chain methods, as they would result from transitioning to macrocanonical partition functions depending on a chemical potential. Lastly we demonstrate the relationship of the partition function approach to the Cram\'er-Rao boundary and to information entropies.