Distributed estimation of binary event probabilities via hierarchical Bayes and dual decomposition

In this paper we consider a network of monitors that can count the occurrences of binary events of interest. The aim is to estimate both the local event probabilities and some global features of the system as, e.g., the mean probability. This scenario is motivated by several applications in cyber-physical systems and social networks. We propose a hierarchical Bayesian approach in which the individual event probabilities are treated as random variables with an \emph{a priori} density function. Following the empirical Bayes approach, the prior is chosen in a family of distributions parameterized by suitable unknown hyperparameters. We develop a distributed optimization algorithm, as a variant of a standard distributed dual decomposition scheme, to obtain locally the Maximum Likelihood estimates of the hyperparameters. These estimates allow each monitor to gain accuracy in both the local and global estimation tasks. This approach is particularly well suited in scenarios in which the number of samples at each node are allowed to be highly inhomogeneous.