  Although Bayesian methods are robust and principled, their application in
practice could be limited since they typically rely on computationally
intensive Markov Chain Monte Carlo algorithms for their implementation. One
possible solution is to find a fast approximation of posterior distribution and
use it for statistical inference. For commonly used approximation methods, such
as Laplace and variational free energy, the objective is mainly defined in
terms of computational convenience as opposed to a true distance measure
between the target and approximating distributions. In this paper, we provide a
geometric view of posterior approximation based on a valid distance measure
derived from ambient Fisher geometry. Our proposed framework is easily
generalizable and can inspire a new class of methods for approximate Bayesian
inference.
