Symplectic/Contact Geometry Related to Bayesian Statistics

Atsuhide Mori

doi:10.3390/ecea-5-06665

Abstract

In the previous work, the author gave the following symplectic/contact geometric description of the Bayesian inference of normal means: The space H of normal distributions is an upper halfplane which admits two operations, namely, the convolution product and the normalized pointwise product of two probability density functions. 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. The graph of F is Lagrangian with respect to the negative symplectic structure. It is contained in the bi-contact hypersurface N. Further, it is preserved under a bi-contact Hamiltonian flow with respect to a single function. Then the restriction of the flow to the graph of F presents the inference of means. The author showed that this also works for the Student t-inference of smoothly moving means and enables us to consider the smoothness of data smoothing. In this presentation, the space of multivariate normal distributions is foliated by means of the Cholesky decomposition of the covariance matrix. This provides a pair of regular Poisson structures, and generalizes the above symplectic/contact description to the multivariate case. The most of the ideas presented here have been described at length in a later article of the author.

Highlights

We work in the C ∞ -smooth category
Since the pointwise product of two normal densities is proportional to a normal density, it induces another product · on H, which we call the Bayesian product
The first half of this presentation is devoted to the geometric description of Bayesian statistics including this product

Summary

Introduction

We work in the C ∞ -smooth category. A manifold U embedded in the space of probability distributions inherits a separating premetric D : U × U → R≥0 from the relative entropy, which is called the Kullback–Leibler divergence. X is the contact Hamiltonian vector field of λ− | N with respect to the same function. Hamiltonian vector field on ( N, λ± ) which is tangent to the leaf F0 It is the one for the above function m up to constant multiple. H × H since F0 is the graph of a diffeomorphism This vector field is tangent to a foliation by e-geodesics, and each leaf is closed under the Bayesian product. The author [4] showed that a similar vector field on the squared space of Student’s t-distributions can provide an indication of “geometric smoothness” in actual data smoothing. The leaves are 4n-dimensional submanifolds carrying two symplectic structures They form a pair of Poisson structures on the squared space. For the precise descriptions and the proofs, see the article [5] by the author

Bayesian Information Geometry

The Geometry of Normal Distributions

Discussion