The Hierarchical Particle Filter
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1 and Arnaud Doucet MCMSki V Lenzerheide 7th January 2016
2 Context & Outline Filtering in State-Space Models: SIR Particle Filters [GSS93] Block-Sampling Particle Filters [DBS06] Exact Approximation of Monte Carlo Algorithms: Particle MCMC [ADH10] and SMC 2 [CJP13] Exact Approximation and Particle Filters: Approximated RBPFs [CSOL11] Exactly [JWD12] Hierarchical SMC [Here] Pseudomarginal State Augmentation: More of the SAME? [Axel Finke; this session]
3 Hidden Markov Models Consider: x 1 x 2 x 3 x 4 x 5 x 6 y 1 y 2 y 3 y 4 y 5 y 6 Unobserved Markov chain {X n } transition f. Observed process {Y n } conditional density g. Density: n p(x 1:n, y 1:n ) = f 1 (x 1 )g(y 1 x 1 ) f(x i x i 1 )g(y i x i ). i=2
4 Sequential Importance Resampling Algorithmically, at iteration n: Given {W i n 1, X i 1:n 1} for i = 1,..., N: Resample, obtaining {1/N, X i 1:n 1}. Sample X i n q n( X i n 1) Weight W i n f(xi n X i n 1 )g(yn Xi n ) q n(x i n X i n 1 ) Actually: Resample efficiently. Only resample when necessary.
5 Particle MCMC MCMC algorithms which employ SMC proposals [ADH10] SMC algorithm as a collection of RVs Values Weights Ancestral Lines Construct MCMC algorithms: With many auxiliary variables Exactly invariant for distribution on extended space Standard MCMC arguments justify strategy SMC 2 employs the same approach within an SMC setting. What else does this allow us to do with SMC?
6 Ancestral Trees t=1 t=2 t=3 a 1 3 =1 a 4 3 =3 a 1 2 =1 a 4 2 =3 b 2 3,1:3 =(1, 1, 2) b 4 3,1:3 =(3, 3, 4) b 6 3,1:3 =(4, 5, 6)
7 SMC Distributions Formally gives rise to the SMC Distribution: ( ) an L+2:n, x n L+1:n, k; x n L = ψ M n,l [ M i=1 ] q(x i n L+1 xn L ) n p=n L+2 [ M ( r(a p w p 1 ) q x i p x ai p p 1) ] r(k w n ) and the conditional SMC Distribution: ( ) ψ n,l M ã k n L+2:n, x k n L+1:n ; x n L b k n L+1:n 1, k, x k n L+1:n = ( q x b k n,n L+1 n L+1 x n L r p=n L+2 i=1 ψn,l M (ã n L+2:n, x n L+1:n, k; x n L ) ) [ n ) ( ( bk n,p w p 1 q x b k n,p p x b n n,p 1 p 1 ) ] r(k w n )
8 Block Sampling: An Idealised Approach At time n, given x 1:n 1 ; discard x n L+1:n 1 : Sample from q(x n L+1:n x n L, y n L+1:n ). Weight with W (x 1:n ) = Optimally, p(x 1:n y 1:n ) p(x 1:n L y 1:n 1 )q(x n L+1:n x n L, y 1:n L+1:n ) q(x n L+1:n x n L, y n L+1:n ) =p(x n L+1:n x n L, y n L+1:n ) W (x 1:n ) p(x 1:n L y 1:n ) p(x 1:n L y 1:n 1 ) =p(y n x 1:n L, y n L+1:n 1 ) Typically intractable; auxiliary variable approach in [DBS06].
9 Local Particle Filtering: Current Trajectories 4 Starting Point n
10 Local Particle Filtering: PF Proposal 4 PF Step n
11 Local Particle Filtering: CPF Auxiliary Proposal 4 CPF Step n
12 Local SMC: Version 1 Not just a Random Weight Particle Filter. Propose from: U n 1 1:M (b 1:n 2, k)p(x 1:n 1 y 1:n 1 )ψn,l(a M n L+2:n, x n L+1:n, k; x n L ) ψ n 1,L 1 M (ã k n L+2:n 1, x k n L+1:n 1 ; x ) n L b n L+2:n 1, x n L+1:n 1 Target: 1:M (b 1:n L, b k n,n L+1:n 1, k)p(x 1:n L, x b k n L+1:n ( a k n L+2:n, x k n L+1:n ; x n L b k n,n L+1:n, x b U n ψ M n,l ψ M n 1,L 1 (ã n L+2:n 1, x n L+1:n 1, k; x n L ). Weight: Zn L+1:n / Z n L+1:n 1. n,n L+1:n y 1:n) k n,n L+1:n n L+1:n )
13 Proposal: q(x t x t 1, y t ) = f(x t x t 1) Stochastic Volatility Bootstrap Local SMC Model: f(x i x i 1 ) =N ( φx i 1, σ 2) g(y i x i ) =N ( 0, β 2 exp(x i ) ) Top Level: Local SMC proposal. Stratified resampling when ESS< N/2. Local SMC Proposal: Weighting: W (x t 1, x t ) f(x t x t 1 )g(y t x t ) f(x t x t 1 ) Resample residually every iteration. = g(y t x t )
14 SV Exchange Rata Data 5 Exchange Rate Data Observed Values n
15 SV Bootstrap Local SMC: M= N = 100, M = 100 Average Number of Unique Values L = 2 L = 4 L = 6 L = n
16 SV Bootstrap Local SMC: M= N = 100, M = 1000 Average Number of Unique Values L = 2 L = 4 L = 6 L = 10 L = 14 L = n
17 SV Bootstrap Local SMC: M= N=100, M=10,000 Average Number of Unique Values L = 2 L = 4 L = 6 L = 10 L = 14 L = 18 L = 22 L = n
18 Local SMC: Version 2 Problems with this PF+CPF scheme: Expensive to run 2 filters per proposal... and large M is required... can t we do better? Using a non-standard CPF/PF proposal is preferable: Running a CPF from time n L + 1 to n, and extending it s paths with a PF step to n + 1, invoking a careful auxiliary variable construction, we reduce computational cost and variance. Leads to importance weights: ˆp(y n L+1:n x bk n 1,n L n L ) ˆp(y n L+1:n 1 x bk n 1,n L n L )
19 Hierarchical Particle Filtering: Current Trajectories 7 Starting Point n
20 Hierarchical Particle Filtering: CPF Proposal Component 7 CPF Step n
21 Hierarchical Particle Filtering: PF Proposal Component 7 PF Step n
22 A Toy Model: Linear Gaussian HMM Linear, Gaussian state transition: f(x t x t 1 ) = N (x t ; x t 1, 1) and likelihood g(y t x t ) = N (y t ; x t, 1) Analytically: Kalman filter/smoother/etc. Simple bootstrap HPF: Local proposal: q(x t x t 1, y t ) = f(x t x t 1 ) Weighting: W (x t 1, x t ) g(y t x t )
23 Bootstrap L rz N e+05 M
24 2 N M e log(rz) L 4 Bootstrap C e
25 In Conclusion SMC can be used hierarchically. Software implementation is not difficult [Joh09, Zho13, Mur13, TCF + 14]. The optimal block-sampling particle filter can be approximated exactly but at a considerable computational cost. Many related things are also possible... EA-RBPFs [JWD12] Tempering in blocks [Joh15] Iterated Particle Filters [GJL15]
26 I [ADH10] [CJP13] [CSOL11] [DBS06] [GJL15] [GSS93] [Joh09] [Joh15] [JWD12] [Mur13] [TCF + 14] [Zho13] C. Andrieu, A. Doucet, and R. Holenstein. Particle Markov chain Monte Carlo. Journal of the Royal Statistical Society B, 72(3): , N. Chopin, P. Jacob, and O. Papaspiliopoulos. SMC 2 : an efficient algorithm for sequential analysis of state space models. Journal of the Royal Statistical Society B, 75(3): , T. Chen, T. Schön, H. Ohlsson, and L. Ljung. Decentralized particle filter with arbitrary state decomposition. IEEE Transactions on Signal Processing, 59(2): , February A. Doucet, M. Briers, and S. Sénécal. Efficient block sampling strategies for sequential Monte Carlo methods. Journal of Computational and Graphical Statistics, 15(3): , P. Guarniero, A. M. Johansen, and A. Lee. The iterated auxiliary particle filter. ArXiv mathematics e-print , ArXiv Mathematics e-prints, N. J. Gordon, S. J. Salmond, and A. F. M. Smith. Novel approach to nonlinear/non-gaussian Bayesian state estimation. IEE Proceedings-F, 140(2): , April A. M. Johansen. SMCTC: Sequential Monte Carlo in C++. Journal of Statistical Software, 30(6):1 41, April A. M. Johansen. On blocks, tempering and particle MCMC for systems identification. In Proceedings of 17th IFAC Symposium on System Identification, Beijing, China., IFAC. To appear. Invited submission. A. M. Johansen, N. Whiteley, and A. Doucet. Exact approximation of Rao-Blackwellised particle filters. In Proceedings of 16th IFAC Symposium on Systems Identification, pages , Brussels, Belgium, July IFAC, IFAC. L. M. Murray. Bayesian state-space modelling on high-performance hardware using LibBi. Mathematics e-print , ArXiv, A. Todeschini, F. Caron, M. Fuentes, P. Legrand, and P. Del Moral. Biips: Software for Bayesian inference with interacting particle systems. Mathematics e-print, ArXiv, Y. Zhou. vsmc: Parallel sequential Monte Carlo in C++. Mathematics e-print , ArXiv, 2013.
27 Key Identity = ψ M n,l (a n L+2:n, x n L+1:n, k; x n L ) p(x n L+1:n x n L, y n L+1:n ) ψ n,l M (a k ( ) [ q x bk n,n L+1 n L+1 x n n L =Ẑn L+1:n/p(y n L+1:n x n L ) n L+2:n, x k r ( ( ) b k n,p w p 1 q p=n L+2 p(x n L+1:n x n L, y n L+1:n ) n L+1:n, k; x n L... ) ) ] x bk n,p r(k w n ) p x bn n,p 1 p 1
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