### Jang, D, Kang, JU, Kruckman, A, Kudo, J and Miller, SJ (2009)

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

Journal of Algebra, Number Theory: Advances and Applications 1(1), pp. 37-60.

**ISSN/ISBN:** Not available at this time.
**DOI:** Not available at this time.

**Abstract:** 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.

**Bibtex:**

```
@article{,
title={Chains of distributions, hierarchical Bayesian models and Benford's Law},
author={Jang, Dennis and Kang, Jung Uk and Kruckman, Alex and Kudo, Jun and Miller, Steven J},
journal={Journal of Algebra, Number Theory: Advances and Applications },
year={2009},
vol={1},
number={1},
pages={37--60},
}
```

**Reference Type:** Journal Article

**Subject Area(s):** Statistics