Information Geometry of Probability Distributions
Investigating the Riemannian structure of statistical manifolds and how Fisher information defines natural distances.
Statistical Manifolds
Information geometry treats families of probability distributions as points on a Riemannian manifold. The Fisher information matrix serves as the metric tensor, defining natural notions of distance and curvature on this space.
Fisher Information as a Metric
Given a parametric family of distributions p(x; θ), the
Fisher information matrix defines the infinitesimal distance between nearby
distributions. This geometric perspective reveals deep connections between
statistical estimation and differential geometry.
Applications to Signal Spaces
In our research, signals from physical systems are modeled as samples from parametric distributions. The Fisher metric on these signal distributions provides a natural measure of how “different” two system states are - a critical capability for structural health monitoring and anomaly detection.
Topics covered