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The problem of reconciling a prior probability law with data was introducedby E. Schr\"odinger in 1931/32. It represents an early formulation of a maximumlikelihood problem. The specific formulation can also be seen as the controlproblem to modify the law of a diffusion process so as to match specificationson marginal distributions at given times. Thereby, in recent years, thisso-called {\em Schr\"odinger Bridge problem} has been at the center of thedevelopment of uncertainty control. However, an unstudied facet of this programhas been to address uncertainty in space and time, modeling the effect of tasksbeing completed, instead of imposing specifications at fixed times. The presentwork is a first study to extend Schr\"odinger's paradigm on such an issue. Itis developed in the context of Markov chains and random walks on graphs.Specifically, we study the case where one marginal distribution represents theinitial state occupation of a Markov chain, while others representfirst-arrival time distributions at absorbing states signifying completion oftasks. We establish that when the prior is Markov, a Markov policy is onceagain optimal with respect to a likelihood cost that follows Schr\"odinger'sdictum.

Schr\"odinger's control and estimation paradigm with spatio-temporal distributions on graphs