Gaussian Processes to Speed up Hamiltonian Monte Carlo

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1 Gaussian Processes to Speed up Hamiltonian Monte Carlo Matthieu Lê Murray, Iain 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, 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. Journal Club 11/04/14 1

2 MCMC Monte Carlo : Rejection sampling, Importance sampling, MCMC : Markov Chain Monte Carlo Sampling technique to estimate a probability distribution Draw samples from complex probability distributions In general : f x P x dx 1 S S s=1 f x s, x (s) ~P(x) Journal Club 11/04/14 2

3 Objective Bayesian Inference : θ : Model parameter x i : Observations P θ : prior on the model parameters P x 1 x n θ : likelyhood, potentially expensive to evaluate P θ x 1 x n = P x 1 x n θ P θ P x 1 x n θ P θ dθ P θ x 1 x n = 1 Z P x 1 x n θ P θ MCMC : sample P θ x 1 x n without the knowledge of Z Journal Club 11/04/14 3

4 Metropolis-Hastings We want to draw samples from P α P 1. Propose new θ from the transition function Q(θ, θ) (e.g. N θ, σ 2 ). The transition from one parameter to another is a Markov Chain 2. Accept the new parameter θ with a probability min (1, P θ Q(θ,θ ) P θ Q(θ,θ) ) Q must be chosen to fulfill some technical requirements. Samples are not independent. Sample N 0,1 ) Journal Club 11/04/14 4

5 Metropolis-Hastings Metropolis-Hastings: 1000 samples, step 0.1 Rejection rate: 44% Problem : The proposed samples come from a Gaussian. There is possibly an important rejection rate. Journal Club 11/04/14 5

6 Hamiltonian Monte Carlo Same idea as Metropolis-Hastings BUT the proposed samples now come from the Hamiltonian dynamics : H = E pot + E kin E pot = log P θ, E kin = p2 2m θ t = E kin p = p m p t = E pot θ θ θ p p Journal Club 11/04/14 6

7 Hamiltonian Monte Carlo Algorithm : Sample p according to its known distribution Run the Hamiltonian dynamics during a time T Accept the new sample with probability : min (1, exp E pot + E pot exp ( E kin + E kin )) 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. Journal Club 11/04/14 7

8 Hamiltonian Monte Carlo Gibbs Sampling Metropolis-Hastings Hamiltonian Monte Carlo Journal Club 11/04/14

9 Hamiltonian Monte Carlo Advantage : The Hamiltonian stays (approximately) constant during the dynamic, hence lower rejection rate! 1000 samples, L = 200, ε =0,01 Rejection rate = 0% 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. Journal Club 11/04/14 9

10 Gaussian Process HMC Same algorithm as HMC BUT the Hamiltonian dynamic is computed using Gaussian process simulating E pot p t = E pot θ Gaussian process = distribution over smooth function to approximate E pot : P E pot θ ~N 0, Σ, Σ pq = ω 0 exp( 1 2 D d=1 x p q 2 2 d x d /ωd ) Journal Club 11/04/14 10

11 Gaussian Process HMC Once the Gaussian process is defined with a covariance matrix, we can predict new values : P E pot θ, E pot, θ ~N μ, σ 2, If the Gaussian process is good, μ(θ ) target density Algorithm : 1. Initialization : Evaluate the target density at D random points to define the Gaussian process. 2. Exploratory phase : HMC with E pot = μ σ : evaluation of points with high target value and high uncertainty. Evaluate the real target density at the end of each iteration. 3. Sampling phase : HMC with E pot = μ. Journal Club 11/04/14 11

12 Gaussian Process HMC 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, Journal Club 11/04/14 12

13 Conclusion 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 Journal Club 11/04/14 13

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