Bayesian Methods in Cosmology

TL;DR

This paper provides a comprehensive overview of Bayesian statistical methods, including theory, computational techniques, and model selection, tailored for astronomers analyzing cosmological data.

Contribution

It offers an accessible introduction to Bayesian inference, covering advanced topics like MCMC, Nested Sampling, and model comparison, specifically applied to cosmology.

Findings

Explains Bayesian inference and its relevance to cosmology.

Details numerical methods like MCMC and Nested Sampling.

Contrasts Bayesian model selection with p-value approaches.

Abstract

These notes aim at presenting an overview of Bayesian statistics, the underlying concepts and application methodology that will be useful to astronomers seeking to analyse and interpret a wide variety of data about the Universe. The level starts from elementary notions, without assuming any previous knowledge of statistical methods, and then progresses to more advanced, research-level topics. After an introduction to the importance of statistical inference for the physical sciences, elementary notions of probability theory and inference are introduced and explained. Bayesian methods are then presented, starting from the meaning of Bayes Theorem and its use as inferential engine, including a discussion on priors and posterior distributions. Numerical methods for generating samples from arbitrary posteriors (including Markov Chain Monte Carlo and Nested Sampling) are then covered. The last section deals with the topic of Bayesian model selection and how it is used to assess the performance of models, and contrasts it with the classical p-value approach. A series of exercises of various levels of difficulty are designed to further the understanding of the theoretical material, including fully worked out solutions for most of them.