Initial marking estimation in labeled Petri nets in a probabilistic setting

Given a labeled Petri net with silent (unobservable) transitions, we are interested in performing initial marking estimation in a probabilistic setting. We assume a known finite set of initial markings, each with some a priori probability, and our goal is to obtain the conditional probabilities of initial markings of the Petri net, conditioned on an observed sequence of labels. Under a Markovian assumption on the probabilistic model, we develop a recursive algorithm that allows us to efficiently determine the conditional probabilities for each possible initial marking (conditioned on the sequence of observations seen so far). We illustrate the proposed methodology via an example and discuss potential applications in the context of initial state opacity for security applications.