Title: How proper are Bayesian models in the astronomical literature?,
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.