  In this study, considering the Fisher information metric (Fisher metric)
given by a specific form, which is the form of weights in statistical physics,
we rewrite the Einstein-Hilbert (EH) action. Then, determining the
transformation rules of the Fisher metric, etc under the coarse-graining, we
perform the coarse-graining toward that rewritten EH action. We finally show an
existence of a trivial fixed-point. Here, the existence of a trivial
fixed-point is not trivial for us because we consider the metric given by the
Fisher metric, which is not the normal metric and has to satisfy some
constraint in the formalism of the the Fisher metric. We use the path-integral
in our analysis. At this time we have to accept that a fundamental constraint
in the formalism of the Fisher metric is broken at the quantum level. However
we consider we can accept this with the thought that some constraints and
causal relations held at the classical level usually get broken at the quantum
level. We finds some problems of the Fisher metric. The space-time we consider
is two-dimensional.
