On the Consistency of Maximum Likelihood Estimators for Causal Network Identification

We consider the problem of identifying parameters from data for systems with dynamics evolving according to a particular class of Markov chain processes, called Bernoulli Autoregressive (BAR) processes. The structure of any BAR model is encoded by a directed graph with p nodes. The edges of the graph indicate causal influences, or equivalently dynamic dependencies. More explicitly, the incoming edges to a node in the graph indicate that the state of the node at a particular time instant, which corresponds to a Bernoulli random variable, is influenced by the states of the corresponding parental nodes in the previous time instant. The associated edge weights determine the corresponding level of influence from each parental node. In the simplest setup, the Bernoulli parameter of a particular node’s state variable is a convex combination of the parental node states in the previous time instant and an additional Bernoulli noise variable; this convex combination corresponds to the associated parental edge weights and the contribution of the local noise variable. In this paper, we focus on the problem of structure and edge weight identification by relying on well-established statistical principles. We present two consistent estimators of the edge weights, a Maximum Likelihood (ML) estimator and a closed-form estimator, and numerically demonstrate that the derived estimators outperform existing algorithms in the literature in terms of sample complexity.