Conditional splitting probabilities for hidden-state inference in drift-diffusive processes

Abstract For a one-dimensional process in a bounded interval, splitting probabilities quantify the likelihood of exiting said interval for the first time via either boundary. For two-dimensional Markov processes {X(t),Y(t)}t∈T, a joint analogue of the splitting probabilities can be defined, which captures the likelihood that the variable X(t), having been initialized at x0∈L, exits L for the first time via either of the interval boundaries and that the variable Y(t), initialized at y0, is yexit at the time of exit. Drawing on Bayes’ theorem, we introduce the related notion of conditional splitting probabilities and formalize a simple scheme that leverages them to partially infer the assumedly hidden state Y(t) from detecting the particle exiting at either interval boundary. We demonstrate the viability of this boundary-based inference scheme for two classes of processes (where X and Y are decoupled, and where they are unidirectionally coupled with X depending on Y) and argue for its broad applicability across fields concerned with the control of driven stochastic processes.