Bayes in the sky: Bayesian inference and model selection in cosmology

TL;DR

This review surveys Bayesian probability theory and its cosmological applications, detailing how Bayes' theorem, priors, and the evidence enable principled parameter inference and model comparison. It explains practical inference via MCMC, describes how the Bayesian evidence penalizes unnecessary model complexity, and covers advances in cosmological parameter extraction including scalability to large data sets and high-dimensional spaces. The work highlights how current data favor the vanilla ${\Lambda}$CDM model while outlining methodological considerations, pitfalls, and future directions for cosmostatistics. Overall, Bayesian methods are positioned as essential for rigorous cosmological analysis and discovery in the data-rich era ahead.

Abstract

The application of Bayesian methods in cosmology and astrophysics has flourished over the past decade, spurred by data sets of increasing size and complexity. In many respects, Bayesian methods have proven to be vastly superior to more traditional statistical tools, offering the advantage of higher efficiency and of a consistent conceptual basis for dealing with the problem of induction in the presence of uncertainty. This trend is likely to continue in the future, when the way we collect, manipulate and analyse observations and compare them with theoretical models will assume an even more central role in cosmology. This review is an introduction to Bayesian methods in cosmology and astrophysics and recent results in the field. I first present Bayesian probability theory and its conceptual underpinnings, Bayes' Theorem and the role of priors. I discuss the problem of parameter inference and its general solution, along with numerical techniques such as Monte Carlo Markov Chain methods. I then review the theory and application of Bayesian model comparison, discussing the notions of Bayesian evidence and effective model complexity, and how to compute and interpret those quantities. Recent developments in cosmological parameter extraction and Bayesian cosmological model building are summarized, highlighting the challenges that lie ahead.

Figures (4)

• Figure 1: The evolution of the B--word: number of articles in astronomy and cosmology with "Bayesian" in the title, as a function of publication year. The number of papers employing one form or another of Bayesian methods is of course much larger than that. Up until about 1995, Bayesian papers were concerned mostly with image reconstruction techniques, while in subsequent years the domain of application grew to include signal processing, parameter extraction, object detection, cosmological model building, decision theory and experiment optimization, and much more. It appears that interest in Bayesian statistics began growing around 2002 (source: NASA/ADS).
• Figure 2: Converging views in Bayesian inference. Two scientists having different prior believes $p({\theta}|I_i)$ about the value of a quantity $\theta$ (panel (a), red and green pdf's) observe one datum with likelihood ${\mathcal{L}}({\theta})$ (panel (b)), after which their posteriors $p(\theta|m_1)$ (panel (c), obtained via Bayes Theorem, Eq. (\ref{['eq:Bayes_Theorem_simple']})) represent their updated states of knowledge on the parameter. After observing 100 data points, the two posteriors have become essentially indistinguishable (d).
• Figure 3: Illustration of Bayesian model comparison for two nested models, where the more complex model has one extra parameter. The outcome of the model comparison depends both on the information content of the data with respect to the a priori available parameter space, $I_{10}$ (horizontal axis) and on the quality of fit (vertical axis, $\lambda$, which gives the number of sigma significance of the measurement for the extra parameter). The contours are computed from Eq. (\ref{['eq:savagedickey']}), assuming a Gaussian likelihood and prior (adapted from Trotta:2005ar).
• Figure 6: Posterior constraints on key cosmological parameters from recent CMB and large scale structure data, compare Table \ref{['tab:cosmo_params']}. Top row, from left to right, posterior pdf (normalized to the peak) for the cosmological constant density in units of the critical density, the (physical) baryons and cold dark matter densities. Bottom row, from left to right: optical depth to reionization, scalar tilt and scalar fluctuations amplitude. Yellow using WMAP 1--yr data, orange WMAP 3--yr data and red adding Sloan Digital Sky Survey galaxy distribution data. Spatial flatness and adiabatic initial conditions have been assumed. This set of only 6 parameters (plus 2 other nuisance parameters not shown here) appear currently sufficient to describe most cosmological observations (adapted from Tegmark:2006az).