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