How proper are Bayesian models in the astronomical literature?

TL;DR

This paper highlights the frequent use of improper priors in astronomical Bayesian studies, which can lead to invalid posteriors and erroneous inferences if not properly checked, emphasizing the importance of prior propriety.

Contribution

It reveals the prevalence of improper prior usage in astronomy literature and discusses the necessity of verifying posterior propriety for valid Bayesian inference.

Findings

30.7% of analyzed articles use improper priors without checking posterior validity

Gibbs MCMC can produce plausible samples even from non-probability posteriors

Importance of establishing scientifically motivated proper priors

Abstract

The well-known Bayes theorem assumes that a posterior distribution is a probability distribution. However, the posterior distribution may no longer be a probability distribution if an improper prior distribution (non-probability measure) such as an unbounded uniform prior is used. Improper priors are often used in the astronomical literature to reflect a lack of prior knowledge, but checking whether the resulting posterior is a probability distribution is sometimes neglected. It turns out that 23 articles out of 75 articles (30.7%) published online in two renowned astronomy journals (ApJ and MNRAS) between Jan 1, 2017 and Oct 15, 2017 make use of Bayesian analyses without rigorously establishing posterior propriety. A disturbing aspect is that a Gibbs-type Markov chain Monte Carlo (MCMC) method can produce a seemingly reasonable posterior sample even when the posterior is not a probability distribution (Hobert and Casella, 1996). In such cases, researchers may erroneously make probabilistic inferences without noticing that the MCMC sample is from a non-existing probability distribution. We review why checking posterior propriety is fundamental in Bayesian analyses, and discuss how to set up scientifically motivated proper priors.