Monte Carlo Approximation of Monte Carlo Filters Adam M. Johansen et al. Collaborators Include: Arnaud Doucet, Axel Finke, Anthony Lee, Nick Whiteley 7th January 2014 Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
Context & Outline Filtering in State-Space Models: ◮ SIR Particle Filters [GSS93] ◮ Rao-Blackwellized Particle Filters [AD02, CL00] ◮ Block-Sampling Particle Filters [DBS06] Exact Approximation of Monte Carlo Algorithms: ◮ Particle MCMC [ADH10] ◮ SMC 2 [CJP13] Approximating the RBPF ◮ Approximated Rao-Blackwellized Particle Filters [CSOL11] ◮ Exactly-approximated RBPFs [JWD12] Approximating the BSPF ◮ Local SMC [JD14] Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
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? Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
Ancestral Trees t=1 t=2 t=3 a 1 a 4 a 1 a 4 3 =1 3 =3 2 =1 2 =3 b 2 b 4 b 6 3 , 1:3 =(1 , 1 , 2) 3 , 1:3 =(3 , 3 , 4) 3 , 1:3 =(4 , 5 , 6) Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
SMC Distributions We’ll need the SMC Distribution : � � ψ M a n − L +2: n , x n − L +1: n , k ; x n − L n,L � M � � �� � � � n � M � a i � x n − L ) q ( x i x i r ( a p | w p − 1 ) p | x p r ( k | w n ) = q n − L +1 p − 1 i =1 p = n − L +2 i =1 and the conditional SMC Distribution : � � � � � � � �� ψ M a ⊖ k x ⊖ k b k x k � n − L +2: n , � n − L +1: n ; x n − L � n − L +1: n − 1 , k, � n − L +1: n n,L ψ M n,L ( � a n − L +2: n , � x n − L +1: n , k ; x n − L ) � � �� = � � � � � n � � � b k b k b n � n,n − L +1 n,p n,p − 1 q x � n − L +1 | x n − L r b k n,p | � q x � | � x r ( k | � w n ) w p − 1 p p − 1 p = n − L +2 Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
A (Rather Broad) Class of Hidden Markov Models y 1 x 1 z 1 x 2 z 2 y 2 x 3 z 3 y 3 ◮ Unobserved Markov chain { ( X n , Z n ) } transition f . ◮ Observed process { Y n } conditional density g . ◮ Density: � n p ( x 1: n , z 1: n , y 1: n ) = f 1 ( x 1 , z 1 ) g ( y 1 | x 1 , z 1 ) f ( x i , z i | x i − 1 , z i − 1 ) g ( y i | x i , z i ) . i =2 Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
Formal Solutions ◮ Filtering and Prediction Recursions: p ( x n , z n | y 1: n − 1 ) g ( y n | x n , z n ) � p ( x n , z n | y 1: n ) = p ( x ′ n , z ′ n | y 1: n − 1 ) g ( y n | x ′ n , z ′ n ) d ( x ′ n , z ′ n ) � p ( x n +1 , z n +1 | y 1: n ) = p ( x n , z n | y 1: n ) f ( x n +1 , z n +1 | x n , z n ) d ( x n , z n ) ◮ Smoothing: p (( x, z ) 1: n | y 1: n ) ∝ p (( x, z ) 1: n − 1 | y 1: n − 1 ) f (( x, z ) n | ( x, z ) n − 1 ) g ( y n | ( x, z ) n ) Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
A Simple SIR Filter Algorithmically, at iteration n : ◮ Given { W i n − 1 , ( X, Z ) i 1: n − 1 } for i = 1 , . . . , N : N , ( � X, � ◮ Resample , obtaining { 1 Z ) i 1: n − 1 } . ◮ Sample ( X, Z ) i n ∼ q n ( ·| ( � X, � Z ) i n − 1 ) n | ( � X, � f (( X,Z ) i Z ) i n − 1 ) g ( y n | ( X,Z ) i n ) ◮ Weight W i n ∝ n | ( � X, � Z ) i q n (( X,Z ) i n − 1 ) Actually: ◮ Resample efficiently. ◮ Only resample when necessary. ◮ . . . Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
A Rao-Blackwellized SIR Filter Algorithmically, at iteration n : ◮ Given { W X,i n − 1 , ( X i 1: n − 1 , p ( z 1: n − 1 | X i 1: n − 1 , y 1: n − 1 ) } ◮ Resample , obtaining { 1 N , ( � 1: n − 1 , p ( z 1: n − 1 | � X i X i 1: n − 1 , y 1: n − 1 )) } . ◮ For i = 1 , . . . , N : n ∼ q n ( ·| � ◮ Sample X i X i n − 1 ) 1: n ← ( � ◮ Set X i X i 1: n − 1 , X i n ) . p ( X i n ,y n | � X i n − 1 ) ◮ Weight W X,i ∝ n n | � q n ( X i X i n − 1 ) ◮ Compute p ( z 1: n | y 1: n , X i 1: n ) . Requires analytically tractable substructure. Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
An Approximate Rao-Blackwellized SIR Filter Algorithmically, at iteration n : ◮ Given { W X,i n − 1 , ( X i p ( z 1: n − 1 | X i 1: n − 1 , � 1: n − 1 , y 1: n − 1 ) } ◮ Resample , obtaining { 1 N , ( � p ( z 1: n − 1 | � X i X i 1: n − 1 , y 1: n − 1 )) } . 1: n − 1 , � ◮ For i = 1 , . . . , N : n ∼ q n ( ·| � ◮ Sample X i X i n − 1 ) 1: n ← ( � ◮ Set X i X i 1: n − 1 , X i n ) . p ( X i n ,y n | � X i � n − 1 ) ◮ Weight W X,i ∝ n n | � q n ( X i X i n − 1 ) ◮ Compute � p ( z 1: n | y 1: n , X i 1: n ) . Is approximate; how does error accumulate? Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
Exactly Approximated Rao-Blackwellized SIR Filter At time n = 1 1 ∼ q x ( ·| y 1 ) . ◮ Sample, X i ∼ q z � � ◮ Sample, Z i,j ·| X i 1 , y 1 . 1 ◮ Compute and normalise the local weights � � 1 , Z i,j w z X i � � 1 , y 1 , Z i,j := p ( X i 1 ) 1 1 1 , Z i,j � , W z,i,j w z X i � � � � := � M 1 1 1 � Z i,j 1 , Z i,k � X i k =1 w z X i q z 1 , y 1 1 1 1 � � M � 1 , y 1 ) := 1 1 , Z i,j p ( X i w z X i define � . 1 1 M j =1 ◮ Compute and normalise the top-level weights � � � � p ( X i w x X i := � 1 , y 1 ) � , W x,i 1 1 w x X i q x � := � � . � N 1 1 1 X i 1 | y 1 k =1 w x X k 1 1 At times n ≥ 2 , resample and do essentially the same again. . . Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
Toy Example: Model We use a simulated sequence of 100 observations from the model defined by the densities: �� x 1 � � 0 � � 1 �� 0 µ ( x 1 , z 1 ) = N ; z 1 0 0 1 �� x n � x n − 1 � 1 � � �� 0 f ( x n , z n | x n − 1 , z n − 1 ) = N ; , z n z n − 1 0 1 � � x n � � σ 2 �� 0 x g ( y n | x n , z n ) = N y n ; , σ 2 z n 0 z Consider IMSE (relative to optimal filter) of filtering estimate of first coordinate marginals. Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
Approximation of the RBPF N = 10 Mean Squared Filtering Error −1 10 N = 20 N = 40 N = 80 −2 10 N = 160 0 1 2 10 10 10 Number of Lower−Level Particles, M For σ 2 x = σ 2 z = 1 . Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
Computational Performance 0 10 N = 10 N = 20 N = 40 Mean Squared Filtering Error N = 80 −1 10 N = 160 −2 10 −3 10 1 2 3 4 5 10 10 10 10 10 Computational Cost, N(M+1) For σ 2 x = σ 2 z = 1 . Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
Computational Performance N = 10 1.9 10 N = 20 N = 40 Mean Squared Filtering Error N = 80 N = 160 1.8 10 1.7 10 1.6 10 1 2 3 4 5 10 10 10 10 10 Computational Cost, N(M+1) x = 10 2 and σ 2 For σ 2 z = 0 . 1 2 . Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
What About Other HMMs / Algorithms? Returning to: 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 Introduction Monte Carlo Approximationof Monte Carlo Filters Approximating the RBPF Adam M. Johansen , Block Sampling Particle Filters References
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