Particle Filtering and Smoothing Methods

Transcription

1 Paricle Filering and Smoohing Mehods Arnaud Douce Deparmen of Saisics, Oxford Universiy Universiy College London 3 rd Ocober 2012 A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

2 Sae-Space Models Le {X } 1 be a laen/hidden X -valued Markov process wih X 1 µ ( ) and X (X 1 = x) f ( x). Le {Y } 1 be an Y-valued Markov observaion process such ha observaions are condiionally independen given {X } 1 and Y (X = x) g ( x). General class of ime series models aka Hidden Markov Models (HMM) including X = Ψ (X 1, V ), Y = Φ (X, W ) where V, W are wo sequences of i.i.d. random variables. Aim: Infer {X } given observaions {Y } on-line or off-line. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

3 Sae-Space Models Sae-space models are ubiquious in conrol, daa mining, economerics, geosciences, sysem biology ec. Since Jan. 2012, more han 13,500 papers have already appeared (source: Google Scholar). Finie Sae-space HMM: X is a finie space, i.e. {X } is a finie Markov chain Y (X = x) g ( x) Linear Gaussian sae-space model X = AX 1 + BV, V i.i.d. N (0, I ) Y = CX + DW, W i.i.d. N (0, I ) A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

4 Sae-Space Models Sochasic Volailiy model X = φx 1 + σv, V i.i.d. N (0, 1) Y = β exp (X /2) W, W i.i.d. N (0, 1) Biochemical Nework model Pr ( X+d 1 =x 1 +1, X+d 2 =x 2 x 1, x 2 ) = α x 1 d + o (d), Pr ( X+d 1 =x 1 1, X+d 2 =x 2 +1 x 1, x 2 ) = β x 1 x 2 d + o (d), Pr ( X+d 1 =x 1, X+d 2 =x 2 1 x 1, x 2 ) = γ x 2 d + o (d), wih Y k = Xk 1 T + W i.i.d. k wih W k N ( 0, σ 2). Nonlinear Diffusion model dx = α (X ) d + β (X ) dv, V Brownian moion Y k = γ (X k T ) +W k, W k i.i.d. N ( 0, σ 2). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

5 Inference in Sae-Space Models Given observaions y 1: := (y 1, y 2,..., y ), inference abou X 1: := (X 1,..., X ) relies on he poserior where p (x 1:, y 1: ) = µ (x 1 ) p (y 1: ) = p (x 1: y 1: ) = p (x 1:, y 1: ) p (y 1: ) k=2 f (x k x k 1 ) }{{}}{{} p(x 1: ) p( y 1: x 1: ) p (x 1:, y 1: ) dx 1: k=1 g (y k x k ), When X is finie & linear Gaussian models, {p (x y 1: )} 1 can be compued exacly. For non-linear models, approximaions are required: EKF, UKF, Gaussian sum filers, ec. Approximaions of {p (x y 1: )} T =1 provide approximaion of p (x 1:T y 1:T ). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

6 Mone Carlo Mehods Basics Assume you can generae X (i) 1: p (x 1: y 1: ) where i = 1,..., N hen MC approximaion is p (x 1: y 1: ) = 1 N N δ (i) X (x 1: ) 1: i=1 Inegraion is sraighforward. ϕ (x 1: ) p (x 1: y 1: ) dx 1: ϕ (x 1: ) p ((x 1: ) y 1: ) dx 1: = 1 N N i=1 ϕ Marginalizaion is sraighforward. X (i) 1: p (x k y 1: ) = p (x 1: y 1: ) dx 1:k 1 dx k+1: = 1 N [ ( )] Basic and key propery: V 1 N N i=1 ϕ = X (i) 1: rae of convergence o zero is independen of dim (X ). N δ (i) X (x k ). k i=1 C ( dim(x )) N, i.e. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

7 Mone Carlo Mehods Problem 1: We canno ypically generae exac samples from p (x 1: y 1: ) for non-linear non-gaussian models. Problem 2: Even if we could, algorihms o generae samples from p (x 1: y 1: ) will have a leas complexiy O (). Paricle Mehods solves parially Problems 1 & 2 by breaking he problem of sampling from p (x 1: y 1: ) ino a collecion of simpler subproblems. Firs approximae p (x 1 y 1 ) and p (y 1 ) a ime 1, hen p (x 1:2 y 1:2 ) and p (y 1:2 ) a ime 2 and so on. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

8 Bayesian Recursion on Pah Space We have p (x 1: y 1: ) = p (x 1:, y 1: ) p (y 1: ) where = g (y x ) = g (y x ) f (x x 1 ) p (y y 1: 1 ) predicive p( x 1: y 1: 1 ) {}}{ f (x x 1 ) p (x 1: 1 y 1: 1 ) p (y y 1: 1 ) p (y y 1: 1 ) = g (y x ) p (x 1: y 1: 1 ) dx 1: Predicion-Updae formulaion p (x 1: 1, y 1: 1 ) p (y 1: 1 ) p (x 1: y 1: 1 ) = f (x x 1 ) p (x 1: 1 y 1: 1 ), p (x 1: y 1: ) = g (y x ) p (x 1: y 1: 1 ). p (y y 1: 1 ) A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

9 Mone Carlo Implemenaion of Predicion Sep Assume you have a ime 1 By sampling X (i) hen p (x 1: 1 y 1: 1 ) = 1 N f N δ (i) X (x 1: 1 ). 1: 1 i=1 ( ) x X (i) 1 and seing X (i) 1: = p (x 1: y 1: 1 ) = 1 N N δ (i) (x 1: X ). 1: i=1 ( X (i) 1: 1, X (i) ) Sampling from f (x x 1 ) is usually sraighforward and is feasible even if f (x x 1 ) does no admi any analyical expression; e.g. biochemical nework models. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

10 Imporance Sampling Implemenaion of Updaing Sep Our arge a ime is p (x 1: y 1: ) = g (y x ) p (x 1: y 1: 1 ) p (y y 1: 1 ) so by subsiuing p (x 1: y 1: 1 ) o p (x 1: y 1: 1 ) we obain p (y y 1: 1 ) = g (y x ) p (x 1: y 1: 1 ) dx 1: We now have = 1 N N ( g y X (i) ). i=1 p (x 1: y 1: ) = g (y x ) p (x 1: y 1: 1 ) = p (y y 1: 1 ) ( wih W (i) g y X (i) ), N i=1 W (i) = 1. N i=1 W (i) δ (i) (x 1: X ). 1: A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

11 Mulinomial Resampling We have a weighed approximaion p (x 1: y 1: ) of p (x 1: y 1: ) p (x 1: y 1: ) = N i=1 W (i) δ (i) (x 1: X ). 1: To obain N samples X (i) 1: approximaely from p (x 1: y 1: ), resample N imes wih replacemen o obain N (i) X (i) 1: p (x 1: y 1: ) N δ (i) X (x 1: ) = 1: i=1 N i=1 p (x 1: y 1: ) = 1 N { } where follow a mulinomial of param. N, This can be achieved in O (N). N (i) N δ X (i) (x 1: ) 1: { W (i) A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46 }.

12 Vanilla Paricle Filer A ime = 1 Sample X (i) 1 µ (x 1 ) hen p (x 1 y 1 ) = N W (i) 1 δ (i) (x 1 X ), W (i) 1 g 1 i=1 ( y 1 X (i) ) 1. Resample X (i) 1 p (x 1 y 1 ) o obain p (x 1 y 1 ) = 1 N N i=1 δ (i) X (x 1 ). 1 A ime 2 Sample X (i) f p (x 1: y 1: ) = ( ) x X (i) 1, se X (i) ( 1: = X (i) 1: 1, X (i) ) and N i=1 ( W (i) δ (i) (x 1: X ), W (i) g 1: Resample X (i) 1: p (x 1: y 1: ) o obain p (x 1: y 1: ) = 1 N N δ (i) X (x 1: ). 1: i=1 y X (i) ). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

13 Paricle Esimaes A ime, we ge p (x 1: y 1: ) = 1 N N δ (i) X (x 1: ). 1: i=1 The marginal likelihood esimae is given by ( 1 p (y 1: ) = p (y k y 1:k 1 ) = N k=1 k=1 N ( g i=1 y k X (i) ) ) k. Compuaional complexiy is O (N) a each ime sep and memory requiremens O (N). If we are only ineresed in p (x y 1: ) or p (s (x 1: ) y 1: ) where s (x 1: ) = Ψ (x, s 1 (x 1: 1 )) - e.g. s (x 1: ) = k=1 x 2 k - is fixed-dimensional hen memory requiremens O (N). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

14 Some Convergence Resuls Numerous convergence resuls are available; see (Del Moral, 2004, Del Moral, D. & Singh, 2013). Le ϕ : X R and consider ϕ = ϕ (x 1: ) p (x 1: y 1: ) dx 1:, ϕ = ϕ (x 1: ) p (x 1: y 1: ) dx 1: = 1 N N ( ϕ i=1 X (i) 1: We can prove ha for any bounded funcion ϕ and any p 1 E [ ϕ ϕ p ] 1/p B () c (p) ϕ, N lim N ( ϕ ϕ N ) N ( 0, σ 2 ). Very weak resuls: For a pah-dependen ϕ (x 1: ), B () and σ 2 ypically increase wih. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46 ).

15 sae sae Figure: p ( x 1 y 1 ) and Ê [ X 1 y 1 ] (op) and paricle approximaion of p ( x 1 y 1 ) A. (boom) Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46 Paricles on Pah-Space - figures by O. Cappė ime index ime index

16 sae sae ime index ime index Figure: p ( x 1 y 1 ), p ( x 2 y 1:2 )and Ê [ X 1 y 1 ], Ê [ X 2 y 1:2 ] (op) and paricle approximaion of p ( x 1:2 y 1:2 ) (boom) A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

17 sae sae ime index ime index Figure: p ( x y 1: ) and Ê [ X y 1: ] for = 1, 2, 3 (op) and paricle approximaion of p ( x 1:3 y 1:3 ) (boom) A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

18 sae sae ime index ime index Figure: p ( x y 1: ) and Ê [ X y 1: ] for = 1,..., 10 (op) and paricle approximaion of p ( x 1:10 y 1:10 ) (boom) A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

19 sae sae ime index ime index Figure: p ( x y 1: ) and Ê [ X y 1: ] for = 1,..., 24 (op) and paricle approximaion of p ( x 1:24 y 1:24 ) (boom) A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

20 Remarks Empirically his paricle mehod provides good approximaions of he marginals {p (x y 1: )} 1. This is wha is only necessary in many applicaions hankfully. The join disribuion p (x 1: y 1: ) is poorly esimaed when is large; i.e. we have in he previous example p (x 1:11 y 1:24 ) = δ X 1:11 (x 1:11 ). Degeneracy problem. For any N and any k, here exiss (k, N) such ha for any (k, N) p (x 1:k y 1: ) = δ X 1:k (x 1:k ) ; p (x 1: y 1: ) is an unreliable approximaion of p (x 1: y 1: ) as. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

21 Anoher Illusraion of he Degeneracy Phenomenon For he linear Gaussian model, we can compue exacly S / where ( ) S = xk 2 p (x 1: y 1: ) dx 1: k=1 using Kalman echniques. We compue he paricle esimae of his quaniy using Ŝ / where ( ) Ŝ = xk 2 p (x 1: y 1: ) dx 1: k=1 can be compued sequenially. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

22 Anoher Illusraion of he Degeneracy Phenomenon Figure: S / obained hrough he Kalman smooher (blue) and is paricle esimae Ŝ / (red). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

23 Sronger Convergence Resuls Assume he following exponenially sabiliy assumpion: For any x 1, x 1 1 p (x y 2:, X 1 = x 1 ) p ( x y 2:, X 1 = x ) 1 dx α for 0 α < 1. 2 Marginal disribuion. For ϕ (x 1: ) = ϕ (x L: ), here exiss B 1, B 2 < s.. E [ ϕ ϕ p ] 1/p B 1 c (p) ϕ N, lim N N ( ϕ ϕ ) N ( 0, σ 2 ) where σ 2 B 2, i.e. here is no accumulaion of numerical errors over ime. Relaive Variance Bound. There exiss B 3 < for no oo large. ( ) ) ( p (y1: ) 2 E p (y 1: ) 1 B 3 N A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

24 Summary Paricle mehods provide consisen esimaes under weak assumpions. Under sabiliy assumpions, we have uniform in ime sabiliy of { p (x y 1: )} 1 and relaive variance of { p (y 1: )} 1 only increases linearly wih. Even under sabiliy assumpions, one does no have uniform in ime sabiliy for { p (x 1: y 1: )} 1. Is i possible o eliminae and/or miigae he degeneracy problem? A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

25 Beer Resampling Schemes { Resampling selecs inegers N i=1 N (i) W (i) δ (i) (x 1: X ) 1: ( ) N (i) } such ha N i=1 Mulinomial Resampling. E = NW (i), ( ) ( ) V N (i) = NW (i) 1 W (i). Residual Resampling. Se Ñ (i) = NW (i) ( mulinomial of parameers N, W (1:N ) ) where N (i) N δ X (i) (x 1: ) 1: { }, sample N i from a W (i) W (i) N 1 Ñ (i) hen se N (i) = Ñ (i) + N (i). Sysemaic Resampling. Sample U 1 U ( 0, 1 ) N and le U i = U 1 + i 1 for i = 2,..., N, hen { N i = N U j : i 1 k=1 W (k) U j i k=1 W (k) A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46 }.

26 Dynamic Resampling To measure he variaion of he weighs, we can use he Effecive Sample Size (ESS) We have ESS = N if W (i) ( N ( ESS = i=1 W (i) ) 2 ) 1 = 1/N for any i and ESS = 1 if W (i) = 1 and W (j) = 0 for j = i. Dynamic Resampling: If he variaion of he weighs as measured by ESS is oo high, e.g. ESS < N/2, hen resample he paricles (Liu & Chen, 1995). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

27 Improving he Sampling Sep Boosrap filer. Sample paricles blindly according o he prior wihou aking ino accoun he observaion Very ineffi cien for vague prior/peaky likelihood. Opimal proposal/perfec adapaion. Implemen he following alernaive updae-propagae Bayesian recursion where Updae p (x 1: 1 y 1: ) = p( y x 1 )p( x 1: 1 y 1: 1 ) p( y y 1: 1 ) Propagae p (x 1: y 1: ) = p (x 1: 1 y 1: ) p (x y, x 1 ) p (x y, x 1 ) = f (x x 1 ) g (y x 1 ) p (y x 1 ) Much more effi cien when applicable; e.g. f (x x 1 ) = N (x ; ϕ (x 1 ), Σ v ), g (y x ) = N (y ; x, Σ w ). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

28 A General Recursion Inroduce an arbirary proposal disribuion q (x y, x 1 ); i.e. an approximaion o p (x y, x 1 ). We have seen ha so clearly where p (x 1: y 1: ) = g (y x ) f (x x 1 ) p (x 1: 1 y 1: 1 ) p (y y 1: 1 ) p (x 1: y 1: ) = w (x 1, x, y ) q (x y, x 1 ) p (x 1: 1 y 1: 1 ) p (y y 1: 1 ) w (x 1, x, y ) = g (y x ) f (x x 1 ) q (x y, x 1 ) This suggess a more general paricle algorihm. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

29 A General Paricle Algorihm { } Assume we have N weighed paricles W (i) 1, X (i) 1: 1 approximaing p (x 1: 1 y 1: 1 ) hen a ime, Sample X (i) ( ) q x y, X (i) 1, se X (i) ( 1: = X (i) 1: 1, X (i) ) and p (x 1: y 1: ) = N i=1 W (i) W (i) f 1 W (i) δ (i) (x 1: X ), 1: ( ) ( X (i) 1 g y X (i) ) ( ). q y, X (i) X (i) X (i) If ESS< N/2 resample X (i) 1: p (x 1: y 1: ) and se W (i) 1 N o obain p (x 1: y 1: ) = 1 N N i=1 δ (i) X (x 1: ). 1: 1 A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

30 Building Proposals Our aim is o selec q (x y, x 1 ) as close as possible o p (x y, x 1 ) as his minimizes he variance of w (x 1, x, y ) = g (y x ) f (x x 1 ). q (x y, x 1 ) Any sandard subopimal filering mehod can be used o approximae p (x y, x 1 ) and p (y x ). Example - Local linearisaion proposal: Le X = ϕ (X 1 ) + V, Y = Ψ (X ) + W, wih V N (0, Σ v ), W N (0, Σ w ). We perform local linearizaion Ψ (x) Y Ψ (ϕ (X 1 )) + (X ϕ (X 1 )) + W x and use as a proposal. ϕ(x 1 ) q (x y, x 1 ) ĝ (y x ) f (x x 1 ). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

31 Block Sampling Paricle Filer Problem: we only sample X a ime so, even if you use p (x y, x 1 ), paricles esimaes could have high variance if V p( x 1 y 1: 1 ) [p (y X 1 )] is high. Block sampling idea: allows yourself o sample again X L+1: 1 as well as X in ligh of y { p( x 1: L y 1: 1 ) }} { p (y y L+1: 1, x L ) p (x 1: 1 y 1: 1 ) dx L+1: 1 p (x 1: L y 1: ) = p (y y L+1: 1 ), p (x 1: y 1: ) = p (x L+1: y L+1:, x L ) p (x 1: L y 1: ). When p (x L+1: y L+1:, x L ) and p (y y L+1: 1, x L ) are no available, one can use approximaions (D., Briers & Senecal, 2006, Whieley & Lee, 2012). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

32 Block Sampling Proposals Variance of incremenal weigh p (y jy p ( x1: L j y1: 1 ). A. Douce (UCL Maserclass Oc. 2012) L +1: 1, x L ) w.r.. 3 rd Ocober / 46

33 Fighing Degeneracy Using MCMC Seps The design of good proposals can be complicaed and/or ime consuming. A sandard and generic way o limi parially degeneracy is known as he Resample-Move algorihm (Gilks & Berzuini, 2001); i.e. using MCMC kernels as a principled way o jier he paricle locaions. A MCMC kernel K (x 1: x 1:) of invarian disribuion p (x 1: y 1: ) is a Markov ransiion kernel wih he propery ha p ( x 1: ) ( ) y 1: = p (x 1: y 1: ) K x 1: x 1: dx1:, i.e. if X 1: p (x 1: y 1: ) and X 1: X 1: K (x 1: X 1:) hen marginally X 1: p (x 1: y 1: ). Conrary o MCMC, we ypically do no use ergodic kernels as on-line mehods are required. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

34 Example: Bearings-only-racking Targe modelled using a sandard consan velociy model X = AX 1 + V i.i.d. where V N (0, Σ). The sae vecor X = ( X 1 X 2 X 3 X 4 ) T conains locaion and velociy componens. One only receives observaions of he bearings of he arge ( ) X Y = an 1 3 X 1 + W where W i.i.d. N ( 0, 10 4) ; i.e. he observaions are almos noiseless. We compare Boosrap filer, Paricle-EKF wih L = 5, 10, MCMC moves L = 10 using dynamic resampling. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

35 Degeneracy for Various Proposals Boosrap RMFL(10) EKF(5) EKF(10) Figure: Average number of unique paricles X (i ) approximaing p ( x y 1:100 ); ime on x-axis, average number of unique paricles on y-axis. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

36 Summary Paricle mehods provide consisen esimaes under weak assumpions. We can esimae {p (x y 1: )} 1 saisfacorily bu our approximaions of {p (x 1: y 1: )} 1 degeneraes as increases because of resampling seps. We can miigae bu no eliminae he degeneracy problem by he design of clever proposals. Smoohing mehods o esimae p (x 1:T y 1:T ) can come o he rescue. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

37 Smoohing in Sae-Space Models Smoohing problem: given a fixed ime T, we are ineresed in p (x 1:T y 1:T ) or some of is marginals, e.g. {p (x y 1:T )} T =1. Smoohing is crucial o parameer esimaion. Direc SMC approximaions of p (x 1:T y 1:T ) and is marginals p (x k y 1:T ) are poor if T is large. SMC provide good approximaions of marginals {p (x y 1: )} 1. This can be used o develop effi cien smoohing esimaes. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

38 Fixed-Lag Smoohing The fixed-lag smoohing approximaion relies on p (x y 1:T ) p (x y 1:+ ) for large enough. and quaniaive bounds can be esablished under sabiliy assumpions. This can be exploied by SMC mehods (Kiagawa & Sao, 2001) { } Algorihmically: sop resampling beyond ime + (Kiagawa & Sao, 2001). X (i) Compuaional cos is O (N) bu non-vanishing bias as N (Olsson & al., 2008). Picking is diffi cul: oo small resuls in p (x y 1:+ ) being a poor approximaion of p (x y 1:T ). oo large improves he approximaion bu degeneracy creeps in. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

39 Forward Filering Backward Smoohing Assume you wan o compue he marginal smoohing disribuions {p (x y 1:T )} T =1 insead of sampling from hem. Forward filering Backward smoohing (FFBS). smooher a {}}{ p (x y 1:T ) = = p (x, x +1 y 1:T ) dx +1 p (x +1 y 1:T ) p (x y 1:, x +1 ) dx +1 = filer a smooher a +1 {}}{{}}{ f (x +1 x ) p (x y 1: ) p (x +1 y 1:T ) dx +1. p (x +1 y 1: ) }{{} backward ransiion p( x y 1:,x +1 ) Condiioned upon y 1:T, {X } T =1 is a backward Markov chain of iniial disribuion p (x T y 1:T ) and inhomogeneous Markov ransiions {p (x y 1:, x +1 )} T 1 =1. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

40 Paricle Forward Filering Backward Smoohing Forward filering: compue and sore { p (x y 1: )} T =1 using your favourie PF. Backward smoohing: For = T 1,..., 1, we have p (x y 1:T ) = N i=1 W (i) T δ X (i) (x ) wih W (i) = 1/N and T T p (x y 1:T ) = p (x y 1: ) }{{} where 1 N N i=1 δ X (i) (x ) = N i=1 W (i) T δ X (i) (x ) W (i) N T = W (j) +1 T j=1 p (x +1 y 1:T ) }{{} N j=1 W (j) +1 T δ X (j) (x +1 ) +1 ( f N l=1 f X (j) +1 f ( x +1 x ) f ( x+1 x ) p( x y 1: )dx dx +1 ( X (j) +1 ) (i) X X (l) Compuaional complexiy is O ( TN 2) bu sampling approximaion in O (TN) (Douc e al., 2011). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46 ).

41 Two-Filer Smoohing An alernaive o FB smoohing is he Two-Filer (TF) formula p (x, x +1 y 1:T ) forward filer backward filer {}}{{}}{ p (x y 1: )f (x +1 x ) p (y +1:T x +1 ) The backward informaion filer saisfies p (y T x T ) = g (y T x T ) and p (y :T x ) = g (y x ) p (y +1:T x +1 ) f (x +1 x ) dx +1 Various paricle mehods have been proposed o approximae {p (y :T x )} T =1 bu rely implicily on p (y :T x ) dx < and ry o come up wih a backward dynamics; e.g. solve X +1 = ϕ (X, V +1 ) X = ϕ 1 (X, V +1 ). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

42 Generalized Two-Filer Smoohing Generalized Two-Filer smoohing (Briers, D. & Maskell, 2010) p (x, x +1 y 1:T ) forward filer backward filer {}}{{}}{ p (x y 1: )f (x +1 x ) p (x +1 y +1:T ) p (x +1 ) }{{} arificial prior where p (x +1 y +1:T ) p (y +1:T x +1 ) p (x +1 ). By consrucion, we now have inegrable p (x +1 y +1:T ) which we can approximae using a backward SMC algorihm argeing {p (x +1:T y +1:T )} 1 =T where p (x :T y :T ) p (x ) T k=+1 f (x k x k 1 ) T k= g (y k x k ). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

43 Paricle Generalized Two-Filer Smoohing Forward filer: compue and sore { p (x y 1: )} T =1 using your favourie PF. Backward filer: compue and sore { p (x y :T ) } T using your =1 favourie PF. Combinaion sep: for any {1,..., T } we have p (x, x +1 y 1:T ) p (x y 1:T ) f (x +1 x ) p (x +1 y +1: ) p (x +1 ( N N f X (j) ) +1 X (i) ( i=1 j=1 p X (j) ) δ (i) X,X (j) (x, x +1 ) Cos O ( N 2 T ) bu O (NT ) hrough imporance sampling (Briers, D. & Singh, 2005; Fearnhead, Wyncoll & Tawn, 2010). A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

44 Comparison Direc Mehod vs Fixed-lag, FB and TF Assume he model is exp. sable and we are ineresed in approximaing ϕ T = ϕ (x ) p (x y 1:T ) dx. Mehod Fixed-lag Direc FB/TF # paricles N N N cos O (TN) O (TN) O ( TN 2),O (TN) Variance O (1/N) O ((T + 1) /N) O (1/N) Bias δ O (1/N) O (1/N) MSE=Bias 2 +Var δ 2 + O (1/N) O ((T + 1) /N) O (1/N) FB/TF provide uniformly good approximaions of {p (x y 1:T )} T =1 whereas direc mehod provide only good approximaion for T small. Fas implemenaions FB and TF of compuaional complexiy O (NT ) ouperform oher approaches in erms of MSE. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

45 Summary Paricle smoohing echniques allow us o solve he degeneracy problem. Paricle fixed-lag smoohing is he simples one bu has non-vanishing bias diffi cul o quanify. Paricle FB and TF algorihms provide uniformly good approximaions of marginal smoohing disribuions conrary o direc mehod. A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46

46 Some References and Resources A.D., J.F.G. De Freias & N.J. Gordon (ediors), Sequenial Mone Carlo Mehods in Pracice, Springer-Verlag: New York, P. Del Moral, Feynman-Kac Formulae: Genealogical and Ineracing Paricle Sysems wih Applicaions, Springer-Verlag: New York, O. Cappé, E. Moulines & T. Ryden, Hidden Markov Models, Springer-Verlag: New York, Webpage wih links o papers and codes: hp:// A. Douce (UCL Maserclass Oc. 2012) 3 rd Ocober / 46