Gaussian Processes to Speed up Hamiltonian Monte Carlo Matthieu Lê Journal Club 11/04/141 Neal, Radford M (2011). " MCMC Using Hamiltonian Dynamics. "

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Presentation transcript:

Gaussian Processes to Speed up Hamiltonian Monte Carlo Matthieu Lê Journal Club 11/04/141 Neal, Radford M (2011). " MCMC Using Hamiltonian Dynamics. " In Steve Brooks, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng. Handbook of Markov Chain Monte Carlo. Chapman and Hall/CRC. Rasmussen, Carl Edward. "Gaussian processes to speed up hybrid Monte Carlo for expensive Bayesian integrals." Bayesian Statistics 7: Proceedings of the 7th Valencia International Meeting. Oxford University Press, Murray, Iain

MCMC Journal Club 11/04/142 Monte Carlo : Rejection sampling, Importance sampling, … MCMC : Markov Chain Monte Carlo Sampling technique to estimate a probability distribution Draw samples from complex probability distributions

Objective Journal Club 11/04/143

Metropolis-Hastings Journal Club 11/04/144

Metropolis-Hastings Journal Club 11/04/145 Problem : The proposed samples come from a Gaussian. There is possibly an important rejection rate. Metropolis-Hastings: 1000 samples, step 0.1 Rejection rate: 44%

Hamiltonian Monte Carlo Journal Club 11/04/146

Hamiltonian Monte Carlo Journal Club 11/04/147 The Energy is conserved so the acceptance probability should theoretically be 1. Because of the numerical precision, we need the Metropolis-Hastings type decision in the end.

Hamiltonian Monte Carlo Journal Club 11/04/14 Gibbs SamplingMetropolis-HastingsHamiltonian Monte Carlo

Journal Club 11/04/149 Advantage : The Hamiltonian stays (approximately) constant during the dynamic, hence lower rejection rate ! Problem : Computing the Hamiltonian dynamic requires computing the model partial derivatives, high number of simulation evaluation ! Neal, Radford M (2011). " MCMC Using Hamiltonian Dynamics. " In Steve Brooks, Andrew Gelman, Galin L. Jones, and Xiao-Li Meng. Handbook of Markov Chain Monte Carlo. Chapman and Hall/CRC.

Gaussian Process HMC Journal Club 11/04/1410

Gaussian Process HMC Journal Club 11/04/1411

Gaussian Process HMC Journal Club 11/04/1412 Rasmussen, Carl Edward. "Gaussian processes to speed up hybrid Monte Carlo for expensive Bayesian integrals." Bayesian Statistics 7: Proceedings of the 7th Valencia International Meeting. Oxford University Press, 2003.

Conclusion Journal Club 11/04/1413 Metropolis-Hastings : few model evaluation per iteration but important rejection rate Hamiltonian Monte Carlo : a lot of model evaluation per iteration but low rejection rate GPHMC : few model evaluation per iteration and low rejection rate BUT : Initialization requires model evaluations to define a “good” Gaussian process BUT : Exploratory phase requires one model evaluation per iteration