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Explore a 48-minute mathematics lecture from the ESI's Thematic Programme on "Infinite-dimensional Geometry: Theory and Applications" that delves into information geometry and its differential geometric framework for studying probability distribution spaces. Learn about the Fisher-Rao metric, a Riemannian metric induced by Fisher information on parametric statistical models, and discover the family of dual alpha-connections. Examine the non-parametric counterparts of both the Fisher-Rao metric and alpha-connections, with particular focus on the Lp-Fisher-Rao metrics - a family of Finsler metrics that generalize the Fisher-Rao metric. Understand how these metrics' geodesics align with alpha-connections on smooth density spaces, the relationship between p and alpha parameters, and why this alignment doesn't persist in probability density spaces. Gain insights into the new variational interpretation of alpha-geodesics as energy minimizing curves.
