Given a labeled Petri net, possibly with silent (unobservable) transitions, we are interested in performing marking estimation in a probabilistic setting. We assume a known initial marking or a known finite set of initial markings, each with some a priori probability, and our goal is to obtain the conditional probabilities of possible markings of the Petri net, conditioned on an observed sequence of labels. Under the assumptions that (i) the set of possible markings, starting from any reachable marking and following any arbitrarily long sequence of unobservable transitions, is bounded, and (ii) a characterization of the a priori probabilities of occurrence for each transition enabled at each reachable marking is available, explicitly or implicitly, we develop a recursive algorithm that efficiently performs current marking estimation.