Outline Introduction Kernel SMC Implementation Details Evaluation Conclusion References Kernel Sequential Monte Carlo Ingmar Schuster * (Paris Dauphine) Heiko Strathmann * (University College London) Brooks Paige (Oxford) Dino Sejdinovic (Oxford) * equal contribution April 25, 2016 1 / 37
37
Embed
Kernel Sequential Monte Carlo - WordPress.com · 11/25/2015 · Intractable likelihoods Intractable Likelihoods and Evidence intractable likelihoods arise in many models (e.g. nonconjugate
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Input: initial proposal density q0, unnormalized density π,population size N, sample size mOutput: lists P,W of m samples and weightsInitialize P = List()Initialize W = List()while len(P) ≤ m do
construct proposal distribution qtgenerate set of p samples Xt from qt and append it to P
for all X ∈ Xt append weights π(X )/qt(X ) to Wend while
Approximate integrals with respect to target distribution πT
Build upon Importance Sampling: approximate integral of hwrt density πT using samples following density q (undercertain conditions):∫
h(x)dπT (x) =
∫h(x)
πT (x)
q(x)dq(x)
Given prior π0, build sequence π0, . . . , πi , . . . πT such that
πi+1 is closer to πT than πi(δ(πi+1, πT ) < δ(πi , πT ) for some divergence δ)sample from πi can approximate πi+1 well usingimportance weight function w(·) = πi+1(·)/πi (·)
Using proposal density q0, generate particles{(w0,j ,X0,j)}Nj=1 where w0,j = π0(X0,j)/q0(X0,j)importance resampling, resulting in Nequally weighted particles {(1/N, X0,j)}Nj=1
rejuvenation move for each X0,j byMarkov Kernel leaving π0 invariant
At i > 0
approximate πi by {(πi (Xi−1,j)/πi−1(Xi−1,j),Xi−1,j)}Nj=1
Fixed learning rate of λ = 0.1 to adapt scale parameter usingstochastic approximation
Geometric bridge of length 20
30 Monte Carlo runs
Report Maximum Mean Discrepancy (MMD) using polynomialkernel of order 3: distance of moments up to order 3 betweenground truth samples and samples produced by each method
Stochastic volatility model with intractable likelihood
Stochastic volatility particularly challenging class of bayesianinverse problems
time series as a high-dimensional nuisance variable
models have to capture the non-linearities in the data(Barndorff-Nielsen and Shephard, 2001)
concentrate on the prediction of daily volatility of asset prices,reusing the model and dataset studied by Chopin et al. (2011)(nuisance of dimension d = 753)
evaluated on several challenging models where it was clearlyimproving statistical efficiency
KASMC exhibits better MMD for Bananaless MC variance than ASMC in evidence estimation for GPclassificationKGRIS clearly improves covariance estimates in StochasticVolatility model
Andrieu, C. and Thoms, J. (2008). A tutorial on adaptive MCMC.Statistics and Computing, 18(November):343–373.
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-GaussianOrnstern-Uhlenbeck-Based Models and Some of Their Uses inFinancial Economics. Journal of the Royal Statistical Society.Series B, 63(2):167–241.
Cappe, O., Guillin, a., Marin, J. M., and Robert, C. P. (2004).Population Monte Carlo. Journal of Computational andGraphical Statistics, 13(4):907–929.
Chopin, N. (2002). A sequential particle filter method for staticmodels. Biometrika, 89(3):539–552.
Chopin, N., Jacob, P. E., and Papaspiliopoulos, O. (2011).SMCˆ2: an efficient algorithm for sequential analysis ofstate-space models. 0(1):1–27.
Fearnhead, P. and Taylor, B. M. (2013). An Adaptive SequentialMonte Carlo Sampler. Bayesian Analysis, (2):411–438.
Rahimi, A. and Recht, B. (2007). Random Features for LargeScale Kernel Machines. In Neural Information ProcessingSystems, number 1, pages 1–8.
Rosenthal, J. S. (2011). Optimal Proposal Distributions andAdaptive MCMC. In Handbook of Markov Chain Monte Carlo,chapter 4, pages 93–112. Chapman & Hall.
Schuster, I. (2015). Consistency of Importance Sampling estimatesbased on dependent sample sets and an application to modelswith factorizing likelihoods. arXiv preprint, pages 1–14.
Sejdinovic, D., Strathmann, H., Garcia, M. L., Andrieu, C., andGretton, A. (2014). Kernel Adaptive Metropolis-Hastings. arXiv,32.
Strathmann, H., Sejdinovic, D., Livingstone, S., Szabo, Z., andGretton, A. (2015). Gradient-free Hamiltonian Monte Carlo withefficient Kernel Exponential Families. In Neural InformationProcessing Systems.
Tran, M.-N., Scharth, M., Pitt, M. K., and Kohn, R. (2013).Importance sampling squared for Bayesian inference in latentvariable models.