The binomial distribution with dependent Bernoulli trials

Abstract

The binomial distribution as the model for the aggregation of independent Bernoulli trials is well known and often used in modeling and data analysis. The generalization of the binomial distribution for dependencies among the Bernoulli trials has received significant attention and several approaches have been suggested to develop computationally feasible solutions. Most of them take the theoretical model of Bahadur–Lazarsfeld for dependent Bernoulli trials as their starting point. The large number of dependency parameters in the Bahadur–Lazarsfeld model, even for a moderate number of trials, has shifted the attention to approximation. However, a computational method that determines the full Bahadur–Lazarsfeld distribution based on the low-order information of marginal expectations and second-order correlations, typically available in applications, is a useful alternative. This article develops such a method using the maximum entropy inference method. An advantage of the method is that improper distributions, which may occur in approximating the Bahadur–Lazarsfeld model by truncation, will never occur. Furthermore, a consistency check can be performed on the user-specified input parameters and, if necessary, restored by a projection of (part of) the input parameters onto the feasible domain. The link between dependent Bernoulli trials and a multi-way binary contingency table renders a characterization of the maximum entropy derived probability model in terms of coefficients of partial association. The method has been implemented and a number of examples show the potential of the method.