  Time-arrow $s=+/-$, intrinsic to a concrete physical system, is associated
with the direction of information loss $\Delta I$ displayed by the random
evolution of the given system. When the information loss tends to zero the
intrinsic time-arrow becomes uncertain. We propose the heuristic relationship
$1/[1+exp(-s\Delta I)]$ for the probability of the intrinsic time-arrow. The
main parts of the present work are trying to confirm this heuristic equation.
The probability of intrinsic time arrow is defined by Bayesian inference from
the observed random process. From irreversible thermodynamic systems, the
proposed heuristic probabilities follow via the Gallavotti-Cohen relations
between time-reversed random processes. In order to explore the underlying
microscopic mechanism, a trivial microscopic process is analyzed and an obvious
discrepancy is identified. It can be resolved by quantum theory. The
corresponding trivial quantum process will exactly confirm the proposed
heuristic time-arrow probability.
