In complex geometry, the Bergman metric plays a very important role as a
canonical metric as a pullback metric of the Fubini-Study metric of complex
projective ambient space. This work is trying to do something really new to
find a whole new approach of studying hyperbolic complex geometry,
especially for a bounded domain in C^n, we replace the infinite dimensional
complex projective ambient space to the collection of probability
distributions defined on a bounded domain. We prove that in this new
framework, the Bergman metric is given as a pullback metric of the
Fisher-Information metric considered in information geometry, and from this,
a new perspective on the contraction property and biholomorphic invariance
of the Bergman metric will be discussed. As an application of this
framework, in the case of bounded hermitian symmetric domains, we will
discuss about the existence of a sequence of i.i.d random variables in which
the covariance matrix converges to a distribution sense with a normal
distribution given by the Bergman metric, and if more time is left, we will
talk about recent progresses on stochastic complex geometry.