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Center for Nonlinear Studies |
The upsurge in application of Bayesian inference comes largely from its computational feasibility made possible by the ability to sample a desired target distribution, with little restriction on the state space. The most popular sampling algorithms are the stochastic, MCMC, samplers that generate a random walk in state space using Metropolis-Hastings dynamics. When Metropolis, et al., proposed their algorithm in the 1950s as a means of simulating a physical system a competing idea was to simulate the system directly and appeal to Liouville's theorem from classical mechanics to ensure that the resulting tour is ergodic -- at least over the region of state space visited. In this talk I look at the possibility of using the latter approach for Bayesian inference, asking whether it is possible to find (deterministic) dynamics on an arbitrary state space that has ergodic properties with respect to the desired target distribution. The current 'hybrid Monte Carlo' algorithms (almost) do this using Hamiltonian dynamics, while more general dynamics can be generated on low-dimensional spaces. Presently it is not clear that a computationally efficient scheme exists for high-dimensional problems, however the interplay between determinism and randomness is fascinating.