The Metalog Distributions: Virtually Unlimited Shape Flexibility, Combining Expert Opinion in Closed Form, and Bayesian Updating in Closed Form

Abstract

Users of probability distributions frequently need to convert data (empirical, simulated, or elicited) into a continuous probability distribution and to update that distribution when new data becomes available. Often, it is unclear which traditional probability distribution(s) to use, fitting to data is laborious and unsatisfactory, little insight emerges, and updating with Bayes rule is impractical. Here we offer an alternative -- a family of continuous probability distributions, fitting methods, and tools that: provide sufficient shape and boundedness flexibility to closely match virtually any probability distribution or data set; involve a single set of simple closed-form equations; stimulate potentially valuable insights when applied to empirical data; are simply fit to data with ordinary least squares; are easy to combine in closed form when weighting the opinion of multiple experts; and, under certain conditions, are easily updated in closed form according to Bayes rule when new data becomes available. The Bayesian updating method is presented in a way that is readily understandable as a fisherman updates his catch probabilities when changing the river on which he fishes. While metalog applications have been shown to improve decision-making, the methods and results herein are broadly applicable to virtually any use of continuous probability in any field of human endeavor. Diverse data sets may be explored and modeled in these new ways with freely available spreadsheets and tools.