Hamiltonian monte carlo on Manifolds Chang liu 2015-09-14
Hamiltonian Monte Carlo on Manifolds Chang Liu 2015-09-14
Outline ntroduction to hamiltonian monte carlo Riemann manifold Langevin and hamiltonian Monte Carlo methods(Girolami Calderhead 2011) Geodesic monte carlo on embedded Manifolds Byrne Girolami, 2013)
Outline • Introduction to Hamiltonian Monte Carlo • Riemann manifold Langevin and Hamiltonian Monte Carlo methods (Girolami & Calderhead, 2011) • Geodesic Monte Carlo on Embedded Manifolds (Byrne & Girolami, 2013)
Introduction to hamiltonian Monte carlo (Ref: Neal, 2011) History Overview Hamiltonian Dynamics MCMC from Hamiltonian dynamics lustrations of hmc and its benefits
Introduction to Hamiltonian Monte Carlo (Ref: Neal, 2011) • History • Overview • Hamiltonian Dynamics • MCMC from Hamiltonian dynamics • Illustrations of HMC and its benefits
History Background: for the task of simulating the distribution of states for a system of idealized molecules (Metropolis, et al., 1953 MCMC with Metropolis test (Alder Wainwright, 1959 ): deterministic approach by simulating Hamiltonian dynamics Birth: combining(hybrid) (Duane, et al., 1987): Hybrid Monte Carlo. Renamed as Hamiltonian monte carlo afterwards Application to statistics (Neal, 1993ab probabilistic inference and Bayesian learning (Neal, 1996a]: neural network models (Ishwaran, 1999 ): generalized linear models
History • Background: for the task of simulating the distribution of states for a system of idealized molecules, – (Metropolis, et al., 1953): MCMC with Metropolis test. – (Alder & Wainwright, 1959): deterministic approach by simulating Hamiltonian dynamics. • Birth: combining (hybrid) – (Duane, et al., 1987): Hybrid Monte Carlo. Renamed as Hamiltonian Monte Carlo afterwards. • Application to statistics – (Neal, 1993ab): probabilistic inference and Bayesian learning – (Neal, 1996a): neural network models – (Ishwaran, 1999): generalized linear models – …
Overview The metropolis-Hastings method for sampling from the target distribution p( by a proposal transition distribution q(xtlxt-1 Sample xt-1 from q(xlxt-1 Set xt=xt-1 with probability 11. pcxi-1dqxt-ikxi-i22 otherwise set xt =xt-1 x t-1t-1 Hamiltonian dynamics can be seen as a proposal generator it provides a deterministic proposal from xt-1 to xt-1 with p(x)invariant. So the acceptance rate is high and the samples are less correlated
Overview • The Metropolis-Hastings method for sampling from the target distribution 𝑝 𝑥 by a proposal transition distribution 𝑞 𝑥𝑡 𝑥𝑡−1 : – Sample 𝑥𝑡−1 ∗ from 𝑞 𝑥 𝑥𝑡−1 – Set 𝑥𝑡 = 𝑥𝑡−1 ∗ with probability min 1, 𝑝 𝑥𝑡−1 ∗ 𝑞 𝑥𝑡−1 𝑥𝑡−1 ∗ 𝑝 𝑥𝑡−1 𝑞 𝑥𝑡−1 ∗ 𝑥𝑡−1 , otherwise set 𝑥𝑡 = 𝑥𝑡−1 • Hamiltonian dynamics can be seen as a proposal generator: it provides a deterministic proposal from 𝑥𝑡−1 to 𝑥𝑡−1 ∗ with 𝑝 𝑥 invariant. So the acceptance rate is high and the samples are less correlated