Probability Density Functions from the Fisher Information Metric

TL;DR

This paper establishes a general relation between the product of probability density functions and the sum of their Fisher information metrics, providing a flexible method for constructing PDFs from Riemannian Fisher metrics.

Contribution

It introduces a novel relation linking probability density functions and Fisher information metrics, enabling new construction methods for PDFs based on Riemannian geometry.

Findings

Derived a relation between product of PDFs and Fisher metric tensors

Developed a method to construct PDFs from arbitrary Fisher information metrics

The construction depends only on continuity and symmetry properties

Abstract

We show a general relation between the spatially disjoint product of probability density functions and the sum of their Fisher information metric tensors. We then utilise this result to give a method for constructing the probability density functions for an arbitrary Riemannian Fisher information metric tensor. We note further that this construction is extremely unconstrained, depending only on certain continuity properties of the probability density functions and a select symmetry of their domains.