A fast, always positive definite and normalizable approximation of non-Gaussian likelihoods

TL;DR

This paper extends the DALI-approximation to handle likelihoods with covariance-dependent parameters, providing a fast, positive definite, and normalizable method that accurately captures non-Gaussian features and outperforms MCMC in speed.

Contribution

The authors develop an improved DALI-approximation that accommodates covariance-dependent parameters, enabling efficient and accurate modeling of complex non-Gaussian likelihoods.

Findings

Successfully reconstructs a ring-shaped likelihood.

At least 1000 times faster than MCMC methods.

Works with severe parameter degeneracies and singular Fisher matrices.

Abstract

In this paper we extent the previously published DALI-approximation for likelihoods to cases in which the parameter dependency is in the covariance matrix. The approximation recovers non-Gaussian likelihoods, and reduces to the Fisher matrix approach in the case of Gaussianity. It works with the minimal assumptions of having Gaussian errors on the data, and a covariance matrix that possesses a converging Taylor approximation. The resulting approximation works in cases of severe parameter degeneracies and in cases where the Fisher matrix is singular. It is at least $1000$ times faster than a typical Monte Carlo Markov Chain run over the same parameter space. Two example applications, to cases of extremely non-Gaussian likelihoods, are presented -- one demonstrates how the method succeeds in reconstructing completely a ring-shaped likelihood. A public code is released here: http://lnasellentin.github.io/DALI/