Circuit Representations of Finite Recurrent Markov Chains

Abstract

In Chapter 2 we have investigated the genesis of finite Markov chains from a collection {𝒞, w _c } of directed circuits and positive numbers. We are now interested in the inverse problem: find a class {𝒞, w _c } of directed circuits (or cycles) c and positive numbers w _c which can describe by either linear or convex expressions the transition probabilities of two finite Markov chains ξ and χ, with reversed parameter-scale and admitting a common invariant probability distribution. The solutions {𝒞, w _c } to this problem will be called the circuit (cycle)representation of ξ. In addition, the class {𝒞, w _c } will be called either “probabilistic” or “deterministic” (“nonrandomized”) according to whether or not the circuits and their weights enjoy or do not enjoy probabilistic interpretations in terms of the chain ξ.