Information transfers and flows in Markov chains as dynamical causal effects

Abstract

A logical sequence of information-theoretic quantifiers of directional (causal) couplings in Markov chains is generated within the framework of dynamical causal effects (DCEs), starting from the simplest DCEs (in terms of localization of their functional elements) and proceeding step-by-step to more complex ones. Thereby, a system of 11 quantifiers is readily obtained, some of them coinciding with previously known causality measures widely used in time series analysis and often called “information transfers” or “flows” (transfer entropy, Ay–Polani information flow, Liang–Kleeman information flow, information response, etc.,) By construction, this step-by-step generation reveals logical relationships between all these quantifiers as specific DCEs. As a further concretization, diverse quantitative relationships between the transfer entropy and the Liang–Kleeman information flow are found both rigorously and numerically for coupled two-state Markov chains.