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