A coarse-grained Markov chain is a hidden Markov model

Abstract

An aggregated or ‘coarse-grained’ Markov chain (in discrete time and on a finite state-space) is a special case of a hidden Markov model, a type of model well known in the statistics literature and extensively applied in a wide variety of time-series or similar analyses. Such a coarse-grained Markov chain does not in general satisfy the Markov property, but recent work appearing in the physics and applied mathematics literature discusses relevant Markov-chain approximations, both at aggregated level and at microstate level, and establishes bounds on the approximation error. This note demonstrates that, for some purposes at least, such approximations are unnecessary; exact properties of hidden Markov models are available and can be used. In particular, the likelihood of the coarse-grained process can be computed routinely, and such a computation makes available various probabilities of interest associated with the process. Furthermore, this likelihood can be maximized numerically if there are unknown parameters and one has data from which one wishes to estimate them. This question of parameter estimation appears not to be addressed in the relevant literature.