Bayes Spaces, An overview

Bayes Hilbert spaces \(B^2(P)\) describe a Hilbert space structure
on a set of mutally continues probability measures, improper priors
and likelihoods with origin \(P\).

The talk will give an overview of Bayes Spaces and their relation to
various concepts of mathematical statistics. Several deep results of
statistics and information theory become obvious and geometrically
intuitive corrolaries, when viewed in the light of this vector space
structure.

The name comes from the fact that vector addition in this space is
given by Bayes theorem. A distribution family is an exponential
family if and only if its a finite dimensional subspaces of a Bayes
Space. In case of regular exponential families its a Bayes Hilbert
Space. The geometry of the space is closely related to Fisher
information. There is a cannocial isometric mapping to \(L^2_0(P)\)
called centered log ratio transform, proving score functions. The
\(P\)-a.s. constant ratio of this centred log ratio transform to the
log density is Kulback Leibler Divergence. The Basis vectors of
conjugated priors can be directly interpreted in terms of
information and the basis of the original family. I.e. we explicitly
give the conjugated prior for every exponential family.

In a multivariate setting, we can identify conditional distributions
with qotient spaces, and provide a straight forward decomposition
into a sum of products of marginal spaces closely related to the
Hammersley Clifford Theorem, Graphical models and generalizing
log-linear models to continues distributions.