Bayesian inference of power law distributions

Observed data from many research disciplines, ranging from cellular biology to economics, often follow a particular long-tailed distribution known as a power law. Despite the ubiquity of natural power laws, inferring the exact form of the distribution from sampled data remains challenging. The possible presence of multiple generative processes giving rise to an unknown weighted mixture of distinct power law distributions in a single dataset presents additional challenges. We present a probabilistic solution to these issues by developing a Bayesian inference approach, with Markov chain Monte Carlo sampling, to accurately estimate power law exponents, the number of mixtures, and their weights, for both discrete and continuous data. We determine an objective prior distribution that is invariant to reparameterization of parameters, and demonstrate its effectiveness to accurately infer exponents, even in the low sample limit. Finally, we provide a comprehensive and documented software package, written in Python, of our Bayesian inference methodology, freely available at https://github.com/AtwalLab/BayesPowerlaw.