Adaptive Percolation Using Subjective Likelihoods

Abstract

A phenomenon that I call “adaptive percolation” commonly arises in biology, business, economics, defense, finance, manufacturing, and the social sciences. Here one wishes to select a handful of entities from a large pool of entities via a process of screening through a hierarchy of sieves. The process is not unlike the percolation of a liquid through a porous medium. The probability model developed here is based on a nested and adaptive Bayesian approach that results in the product of beta-binomial distributions with common parameters. The common parameters happen to be the observed data. I call this the percolated beta-binomial distribution . The model turns out to be a slight generalization of the probabilistic model used in percolation theory. The generalization is a consequence of using a subjectively specified likelihood function to construct a probability model. The notion of using likelihoods for constructing probability models is not a part of the conventional toolkit of applied probabilists. To the best of my knowledge, a use of the product of beta-binomial distributions as a probability model for Bernoulli trials appears to be new. The development of the material of this article is illustrated via data from the 2009 astronaut selection program, which motivated this work.

Keywords:

• Bernoulli trials
• Beta-binomial distribution
• Finance
• Layered defense
• Percolation
• Percolated beta-binomial distribution
• Source selection

JEL Classification:

• C11
• C13
• C19

ACKNOWLEDGMENTS

Gregory Kovacs drew my attention to the astronaut selection data and made several comments to a previous draft of the article. Owen Martin alerted me to the possible relevance of the work described here to the scenario of Collateralized Mortgage Obligations in finance. The comments by an astute referee, and Ehsan Soofi's nagging insistence that I address each and every comment of the referee, and also his comments, has resulted in a sharpened version of this article. Sudip Bose alerted me to the interesting papers of Crowder and Sweeting and Polson and Wasserman. To all go my sincere thanks.

Research supported by the Army Research Office Grant W911NF-09-1-0039, and by the National Science Foundation Grant DMS-0915156, with the George Washington University. This article is dedicated to the memory of Arnold Zellner whose enthusiasm, humanity, and inspiration will be missed.