How proper are Bayesian models in the astronomical literature?

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.