  The transfer entropy is a well-established measure of information flow, which
quantifies directed influence between two stochastic time series and has been
shown to be useful in a variety fields of science. Here we introduce the
transfer entropy of the backward time series called the backward transfer
entropy, and show that the backward transfer entropy quantifies how far it is
from dynamics to a hidden Markov model. Furthermore, we discuss physical
interpretations of the backward transfer entropy in completely different
settings of thermodynamics for information processing and the gambling with
side information. In both settings of thermodynamics and the gambling, the
backward transfer entropy characterizes a possible loss of some benefit, where
the conventional transfer entropy characterizes a possible benefit. Our result
implies the deep connection between thermodynamics and the gambling in the
presence of information flow, and that the backward transfer entropy would be
useful as a novel measure of information flow in nonequilibrium thermodynamics,
biochemical sciences, economics and statistics.
