Symplectic/Contact Geometry Related to Bayesian Statistics

We have the following symplectic/contact geometric description of the Bayesian inference of means: The space H of normal distributions is an upper halfplane with the two operations presenting the convolution and the normalized product of two densities. There is a diffeomorphism F of H that interchanges these operations as well as sends any e-geodesic to an e-geodesic. The product of two copies of H carries positive and negative symplectic structures and a bi-contact hypersurface N naturally generating these structures. Here the graph of F is Lagrangian with respect to the negative symplectic structure. Further, F is contained in the hypersurface N and preserved under a bi-contact flow. Then the restriction of the flow presents the inference of means. This also works for the Student t-inference of moving means and enables us to consider the smoothness of a data smoothing.

In this presentation, we will foliate the space of multivariate normal distributions by using the Cholesky decomposition of the covariance matrix to generalize the above description. Note that Hideyuki Ishi first pointed out the importance of the Cholesky decomposition in the information geometry of normal distributions. We will also construct a Lorenzian metric associated with the relative entropy. The ultimate aim of this research is to construct a relativistic space-time consisting of (tuples of) distributions, since anything can learn by changing its inner distribution in the Bayesian world view.