A BIC‐based consistent metric between Markovian processes

In this paper, we address the problem of deciding if two independent samples coming from discrete Markovian processes are governed by the same stochastic law. We establish a local metric between samples based on the Bayesian information criterion. In addition, we derive the bound that must be used in this metric to take the decision. In the case on which is decided that the laws are not the same, the metric allows to detect the specific elements of the state space where the discrepancies are manifested. We prove that the metric is statistically consistent to detect if the samples follow the same law, tending to zero when the sample sizes increase. Moreover, we show that the metric assumes arbitrarily large values when the sample sizes increase and the stochastic laws are different. This concept is applied to analyze two lines of production of alcohol fuel, described by five variables each. We identify the variables that most contribute to the discrepancy and, using the local nature of the metric, we list the realizations in which the processes behave differently.