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Statistical Inference in Nonlinear Spaces via Maximum Likelihood and Diffusion Bridge Simulation

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GFSW03 - Shape analysis and computational anatomy

Co-authors: Darryl D. Holm (Imperial College London), Alexis Arnaudon (Imperial College London) , Sarang Joshi (University of Utah)

An alternative to performing statistical inference in manifolds by optimizating least squares criterions such as those defining the Frechet mean is to optimize the likelihood of data. This approach emphasizes maximum likelihood means over Frechet means, and it in general allows generalization of Euclidean statistical procedures defined via the data likelihood. While parametric families of probability distributions are generally hard to construct in nonlinear spaces, transition densities of stochastic processes provide a geometrically natural way of defining data likelihoods. Examples of this includes the stochastic EPDiff framework, Riemannian Brownian motions and anisotropic generalizations of the Euclidean normal distribution. In the talk, we discuss likeliood based inference on manifolds and procedures for approximating data likelihood by simulation of manifold and Lie group valued diffusion bridges.

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