Statistics/MLE for MaxEnt-digamma distributions
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MaxEnt statistics for non-exponential distributions

Many statistical distributions, particularly among social and biological systems, have "heavy tails": situations where rare events are not as improbable as might have been guessed from more traditional statistics. Heavy-tailed distributions are the basis for the phrase "the rich get richer". Here, we propose a basic principle underlying systems with heavy-tailed distributions. We show that it is the same principle (maximum entropy) used in statistical physics and statistics to estimate probabilistic models from relatively few constraints. This principle can be expressed in terms of shared costs and economies of scale. The probability distribution we derive is a mathematical digamma function, and we show that it accurately fits 13 real-world data sets.


J. Peterson, P. Dixit, and K. Dill. A maximum entropy framework for nonexponential distributions. Proc. Natl. Acad. Sci. USA 110(51): 20380--20385, 2013.


Probability distributions having power-law tails are observed in a broad range of social, economic, and biological systems. We describe here a potentially useful common framework. We derive distribution functions p_k for situations in which a 'joiner particle' k pays some form of price to enter a `community' of size k-1, where costs are subject to economies-of-scale (EOS). Maximizing the Boltzmann-Gibbs-Shannon entropy subject to this energy-like constraint predicts a distribution having a power-law tail; it reduces to the Boltzmann distribution in the absence of EOS. We show that the predicted function gives excellent fits to 13 different distribution functions, ranging from friendship links in social networks, to protein-protein interactions, to the severity of terrorist attacks. This approach may give useful insights into when to expect power-law distributions in the natural and social sciences.


Please direct software + statistics questions to Jack Peterson (, and MaxEnt + physics questions to Ken Dill (

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