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The transfer entropy is a well-established measure of information flow, whichquantifies directed influence between two stochastic time series and has beenshown to be useful in a variety fields of science. Here we introduce thetransfer entropy of the backward time series called the backward transferentropy, and show that the backward transfer entropy quantifies how far it isfrom dynamics to a hidden Markov model. Furthermore, we discuss physicalinterpretations of the backward transfer entropy in completely differentsettings of thermodynamics for information processing and the gambling withside information. In both settings of thermodynamics and the gambling, thebackward transfer entropy characterizes a possible loss of some benefit, wherethe conventional transfer entropy characterizes a possible benefit. Our resultimplies the deep connection between thermodynamics and the gambling in thepresence of information flow, and that the backward transfer entropy would beuseful as a novel measure of information flow in nonequilibrium thermodynamics,biochemical sciences, economics and statistics.

scientific reports

Backward transfer entropy: Informational measure for detecting hidden  Markov models and its interpretations in thermodynamics, gambling and  causality