Relationship between Schreiber's transfer entropy and Liang--Kleeman information flow from the perspective of stochastic thermodynamics

The Schreiber's transfer entropy is an important index for investigating the causal relationship between random variables. The Liang--Kleeman information flow is another method to demonstrate the causality within dynamical systems. The Horowitz--Esposito information flow is introduced through stochastic thermodynamics. In this study, I elucidate the relationship between the Schreiber's transfer entropy and the Liang--Kleeman information flow through the Horowitz--Esposito information flow. I consider the case where the system changes according to the stochastic differential equation. It is found that the Liang--Kleeman and Horowitz--Esposito information flows differ by a term derived from the stochastic fluctuation. It is shown that the Schreiber's transfer entropy is not less than the Horowitz--Esposito information flow. This study helps to unify various indexes that determine the causal relationship between variables.