Analytic Methods for Cosmological Likelihoods

TL;DR

This paper introduces analytic techniques for cosmological likelihood analysis that efficiently handle multiple parameters, improve computational speed, and facilitate Bayesian model selection.

Contribution

It provides general analytic methods for likelihood maximization, marginalization, and evidence calculation, enhancing efficiency in cosmological data analysis.

Findings

Analytic marginalization preserves information about remaining parameters.

Methods significantly speed up likelihood analysis when combined with MCMC.

Bayesian model selection becomes effectively instantaneous with these techniques.

Abstract

We present general, analytic methods for Cosmological likelihood analysis and solve the "many-parameters" problem in Cosmology. Maxima are found by Newton's Method, while marginalization over nuisance parameters, and parameter errors and covariances are estimated by analytic marginalization of an arbitrary likelihood function with flat or Gaussian priors. We show that information about remaining parameters is preserved by marginalization. Marginalizing over all parameters, we find an analytic expression for the Bayesian evidence for model selection. We apply these methods to data described by Gaussian likelihoods with parameters in the mean and covariance. This method can speed up conventional likelihood analysis by orders of magnitude when combined with Monte-Carlo Markov Chain methods, while Bayesian model selection becomes effectively instantaneous.