  Exponential families and mixture families are parametric probability models
that can be geometrically studied as smooth statistical manifolds with respect
to any statistical divergence like the Kullback-Leibler (KL) divergence or the
Hellinger divergence. When equipping a statistical manifold with the KL
divergence, the induced manifold structure is dually flat, and the KL
divergence between distributions amounts to an equivalent Bregman divergence on
their corresponding parameters. In practice, the corresponding Bregman
generators of mixture/exponential families require to perform definite integral
calculus that can either be too time-consuming (for exponentially large
discrete support case) or even do not admit closed-form formula (for continuous
support case). In these cases, the dually flat construction remains theoretical
and cannot be used by information-geometric algorithms. To bypass this problem,
we consider performing stochastic Monte Carlo (MC) estimation of those
integral-based mixture/exponential family Bregman generators. We show that,
under natural assumptions, these MC generators are almost surely Bregman
generators. We define a series of dually flat information geometries, termed
Monte Carlo Information Geometries, that increasingly-finely approximate the
untractable geometry. The advantage of this MCIG is that it allows a practical
use of the Bregman algorithmic toolbox on a wide range of probability
distribution families. We demonstrate our approach with a clustering task on a
mixture family manifold.
