Decisional states

The computational mechanics de nition of a process internal states considers optimal prediction as the criterion for clustering: the process states consist of the histories that lead to the same distribution of future events. Knowledge of the current cluster for a given past, and its associated distribution of possible futures, is the minimal information necessary for making optimal predictions. However when given a practical problem for which a unique prediction is sought, this view o ers no distinction between futures with rather di erent consequences, incurred costs or expected utility. When each future is ponderated by its would-be e ect then decision theory must be used. This article explores the consequences of grouping histories into states with equivalent optimal decisions in terms of utility, rather than equivalent full distributions of futures. The transitions between these decisional states correspond to events that lead to a change of decision. The utility function encodes the a priori knowledge on a system, the decisional states represents the underlying structure matching both the intrinsic system causal states and the external information. An algorithm is provided so as to estimate the states and their transitions from data. Application examples are given for discrete process hidden state reconstruction, cellular automata ltering, and edge detection in images.