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Kossovsky recently conjectured that the distribution of leading digits of achain of probability distributions converges to Benford's law as the length ofthe chain grows. We prove his conjecture in many cases, and provide aninterpretation in terms of products of independent random variables and acentral limit theorem. An interesting consequence is that in hierarchicalBayesian models priors tend to satisfy Benford's Law as the number of levels ofthe hierarchy increases, which allows us to develop some simple tests (based onBenford's law) to test proposed models. We give explicit formulas for the errorterms as sums of Mellin transforms, which converges extremely rapidly as thenumber of terms in the chain grows. We may interpret our results as showingthat certain Markov chain Monte Carlo processes are rapidly mixing to Benford'slaw.

Chains of distributions, hierarchical Bayesian models and Benford's Law