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27 January 2011 Probability distributions from Riemannian geometry, generalized hybrid Monte Carlo sampling, and path integrals
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Abstract
When considering probabilistic pattern recognition methods, especially methods based on Bayesian analysis, the probabilistic distribution is of the utmost importance. However, despite the fact that the geometry associated with the probability distribution constitutes essential background information, it is often not ascertained. This paper discusses how the standard Euclidian geometry should be generalized to the Riemannian geometry when a curvature is observed in the distribution. To this end, the probability distribution is defined for curved geometry. In order to calculate the probability distribution, a Lagrangian and a Hamiltonian constructed from curvature invariants are associated with the Riemannian geometry and a generalized hybrid Monte Carlo sampling is introduced. Finally, we consider the calculation of the probability distribution and the expectation in Riemannian space with path integrals, which allows a direct extension of the concept of probability to curved space.
© (2011) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
E. Paquet and H. L. Viktor "Probability distributions from Riemannian geometry, generalized hybrid Monte Carlo sampling, and path integrals", Proc. SPIE 7864, Three-Dimensional Imaging, Interaction, and Measurement, 78640X (27 January 2011); https://doi.org/10.1117/12.872862
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