Short Course. Rong Chen Rutgers University Peking University

Transcription

1 Shor Course Sae Space Models, Generalized Dynamic Sysems and Sequenial Mone Carlo Mehods, and heir applicaions in Engineering, Bioinformaics and Finance Rong Chen Rugers Universiy Peking Universiy 1

2 Par Two: Sequenial Mone Carlo Mehods he Framework and Implemenaion.1 A Framework.1.1 (Opional) Inermediae Disribuions.1. Propagaion: Sampling Disribuion.1.3 Resampling/Rejuvenaion.1.4 Inference: Rao-Blackwellizaion. Some Theoreically Resuls.3 Some Applicaions (in deail)

3 .1.3 Resampling (rejuvenaion) Fac: Variance of w increases (sochasically) as increases SMC does no allow o go back o correc early samples Carrying samples wih small weigh forward wases compuaional resources Soluion: duplicae he imporan samples and remove he unimporan samples. 3

4 Simple resampling: A ime, a se of samples S = {(x (j), w (j) )} m j=1 Simple Resampling Sep: (A) Sample a new se of sreams S from S according o w (j), wih replacemen. (B) All sampled samples are assigned weigh 1. The resampled samples behaves as idenical (bu no independen) samples from π (x ). Homework: Show he new sample is sill properly weighed wih respec o π (x ). 4

5 Residual resampling: A ime, a se of samples S = {(x (j) Make mw (j) copies of x (j)., w (j) )} m j=1. m j=1 w(j) = 1. Le m = m m j=1 mw(j) and w (j) = mw (j) mw (j), j = 1,...m. Resample m samples from S wih probabiliy proporional o w (j) wih replacemen. Prune-and-Enriched Rosenbluh Mehod (Grassberger 1997): (Sequenially) Replacing each zero weigh sample wih he sample of highes weigh. The weigh of boh he original sample and he duplicaed sample are se o half of he original weigh. Homework: Show he new sample is sill properly weighed wih respec o π (x ). 5

6 Remarks: Resampling provides more efficien samples of fuure saes Resampling increases sampling variaion in he pas saes Resampling reduces he number of disincive samples in he pas saes Frequen resampling can be shorsighed (online esimaion) Resampling should be done afer esimaion. Resampling Schedule: deerminisic: resampling a ime,, 3,... dynamic: monioring he weigh variance 6

7 A simulaed example: y = x +.8x 1.4x + ε wih x i.i.d from {, 1, 3} and SNR=15dB. The coefficiens φ are inegraed ou wih a normal prior. simulaed sequences. Sample size T =. Number of sreams m = 1. Delayed esimaion: ˆx = MAP (π +3 (x )) simple random sampling (s) versus residual sampling (r) Deerminisic schedule:,, 3,... Dynamic schedule: when he effecive sample size is less han 3. 7

8 Deerminisic Resampling Schedule dynamic schedule error s r s r s r s r s r s r s r >

9 Why resampling? The asympoic variance ( m) (esimaing µ = h(x n )π n (x n )dx n ) No resampling: π n (x n )(h(x n ) µ) dx n g(x n ) Resampling (Del Moral 4, Chopin, 4) where π n (x 1 )(µ 1 (x 1 ) µ) dx 1 + g 1 (x 1 ) µ (x ) = n = π n(x )(µ (x ) µ) π 1 (x 1 )g (x x 1 ) dx h(x )π n (x +1:n x )dx +1:n (a much smooher funcion and closer o µ) e.g. he firs erm: same as sample from g 1 (x 1 )π n (x :n ) 9

10 Flexible Resampling Schemes The resampling rick: Suppose x (1),..., x (m) following g(x ) Sample m samples (wih replacemen) from he se {x (1),..., x (m) } wih probabiliy proporional o α (x (j) ), j = 1,..., m. The resuling se asympoically follow he disribuion e.g. α(x ) = π (x )/g (x ) g(x )α(x ) One can choose differen (and beer) α(x ) o serve differen purposes. 1

11 Flexible Resampling Schemes The square-roo of weighs (Liu, 1) α 1 (x 1 ) = w 1 (x 1 ) Auxiliary paricle filer (Pi & Shephard, 1999) α 1 (x 1 ) = w 1 (x 1 )γ (ˆx x 1 ) where ˆx is a (global) predicion of x. Incremenal-Weigh Spreading. (Neil Shephard, privae conversaion) α 1 (x 1 ) = [ L u l (x l )] 1/L = l=1 L [ π l (x l ) π l 1 (x l 1 )g l (x l x l ) l=1 ] 1/L 11

12 Delayed resampling and block sampling (Wang e al,, Douce e al 6) α 1 (x 1 ) = π +δ (x 1 ) π +δ 1 (x )g (x 1 x 1 ) Resampling wih backward pilos, (Lin e al, 9) α 1 (x 1 ) = ˆπ n (x 1 ) ˆπ n (x )g 1 (x 1 x ) Resampling wih funcion consideraion (Zhang e al, 3) α 1 (x 1 ) = ˆµ 1 (x 1 )w 1 (x 1 ) where ˆµ 1 (x 1 ) is an esimae of h(x n )π n (x n ) dx n 1

13 A imes =,..., n, () Consruc α 1 = {α(x (1) 1 ),..., α(x(m) 1 )} (A) Sample A (j) 1 wih prob {α(1) 1,..., α(m) 1 } (B) sample x (j) g ( x A(j) 1 1 ) and se x(j) := (x A(j) 1 1, x(j) ), and (C) compue and normalize he weighs u (x (j) 1: ) = π (x (j) ) w (x (j) 1: ) = u (x (j) and π 1 (x A(j) 1 1 )g (x (j) (j) n 1 1: )W 1(x A n 1 ) x A(j) 1 = α 1 (x A(j) 1 1 ) W (j) = n 1 ) g 1 (x A(j) 1 1 ) i=1 g i(x (j) i w (x (j) ) m j=1 w (x (j) ) π (x (j) ) x A(j) i 1 i 1 ) 1 i=1 α i(x A(j) i 1 i 1 ) 13

14 Why is i beneficial? In fac, flexible resampling is nohing bu changing he inermediae disribuion. Under flexible resampling scheme, he new inermediae disribuion is 1 π (x ) [g i (x i x i 1 )α i (x i 1 )] i=1 When α (x ) = w (x ) = we ge back π (x ) = π (x ). π (x ) π (x 1 )g (x x 1 ) 14

15 Ofen, here are naural inermediae disribuions. In sae space model, π (x ) = p(x y 1,..., y ). Ofen, he inermediae disribuions guides he design of g (x x 1 ) g (x x 1 ) close o π (x x 1 ) The design of α (x ) can depend on he curren samples of x. Adapiviy. α (x ) = w β (x ) where β depends on he variance of he curren weigh (for example). 15

16 Opimal inermediae disribuion: π (x ) = π n (x ) (he rue marginal) The variance becomes π n (x 1 )(µ 1 (x 1 ) µ) dx 1 + g 1 (x 1 ) n = πn (x x 1 ) g (x x 1 ) π n(x 1 )(µ (x ) µ) dx 1: Almos like each sep is from he rue disribuion π n (x ). 16

17 Delayed resampling: π (x ) = π +δ (x ) wih α (x ) = π +δ (x ) g (x x 1 )α 1 (x 1 ) Or an approximaed delayed resampling π (x ) = ˆπ +δ (x ) 17

18 If α (x ) = ˆπ () n (x ) g (x x 1 )α 1 (x 1 ), = 1,..., n 1 (achievable in cerain cases, e.g. backward pilo) hen π n (x 1 )(µ 1 µ) dx 1 + g 1 (x 1 ) n = π n(x )(µ µ) ˆπ () n (x 1 )g (x x 1 ) dx 1: The difference is beween π 1 (x 1 ) and ˆπ () n (x 1 ) 18

19 Combined sampling and resampling scheme: (discree sae space) If x akes values in {a 1,..., a k } Evaluae α (x (j) 1, a i), i = 1,..., k, j = 1,..., m. Sample m disinc samples from {(x (j) 1, a i), i = 1,..., k, j = 1,..., m} wih probabiliy proporional o α (x (j) 1, a i). Updae weighs 19

20 Applicaion: SALs Saring and Ending a (, ) Inermediae disribuions π (x ): uniform of all SAW such ha d(x ) < n (suppor) where d(x ) = x,1 + x, Combined sampling and resampling Freedom: δ(x ) = n d(x ) Flexibiliy: Prioriy score β(x ) = x,1 x, d(x ) d(x ) (δ(x ) + 1) α (x ) = w 1 exp { [c 1 + δ(x ) c + β(x ]} ). T 1 T wih emperaure sequences T 1 and T.

21 SAL: Log(oal number) # of inside voids (mean) 55 5 Enumerae SMC 4 35 Enumerae SMC

22 Example: SAW wih shape-specific void

23 Le Ω be he se of all lengh-n SAWs. Le C ν be he se of all lengh-n conformaion wih void ν Esimae: P (x n C ν x n Ω) = x n C ν 1 x n Ω 1 Problem: Grow a SAW of lengh-n in C ν One possible soluion: rejecion mehod oo inefficien 3

24 Inermediae disribuions: order of growh Selec he monomers on he void wall firs Then grow he segmens beween he monomers on he wall Sampling disribuion: Self-avoiding Shrinking suppor disance, conneciviy Lookahead Resampling score: freedom and flexibiliy 4

25 5

26 Fracion of conformaions: void 3.1 void x void 4.1 void 4. void 4.3 void 4.4 void lengh (void size=3), lengh (void size=4) 6

27 Fracion of conformaion where ˆf(ν, n) = c 1 r(ν)[(1 c e(ν))c w(ν)+14 3 (n w(ν) + 1) c 4 ] w(ν): wall size e(ν): number of ouer corners r(ν): number of differen roaional ransformaions 7

28 Ou-sample predicion 11 x void 6.1 void 6. void lengh (void size=6) 8

29 Example: Generaing Samples of Diffusion Bridges Generae p(x 1,..., x n 1 x = a, x n = b) Sequenial: p(x x, x n, x 1 ) p(x x 1 )p(x n x ) Use backward pilos o esimae p(x n x ). resampling according o α(x ) = w (x )ˆπ n (x x n ) 35 3 "Pefec" sampling disribuion 14 1 Backward Pilo Forward Sample 5 1 v v (a) f(v k ) (b) k 9

30 Example (Beskos e al. 6) dv = sin(v θ)d + dw Comparison beween exac sampling (Beskos e al. 6), SMC- and SMC-1. 1 realizaions. Sepsize.1. Performance measure L(θ): exac sampling, 1M samples. [ ] 1 1/ 1 RMSE(θ) = ( 1 ˆL i (θ) L(θ)) Observaion sep size = 3 Eular approximaion Roughly same CPU ime i=1 3

31 v /π v /π

32 (Average) Likelihood funcion: 1.5 x θ 3

33 5 "Perfec" sampling disribuion 4 3 v/π v/π v/π 4 5 Sampling disribuion of SMC Sampling disribuion of SMC

34 Esimaion of he likelihood funcion: RM SE exac SMC- SMC-1 m 8, 3,5 1, θ =.π θ =.π θ =.4π θ =.6π θ =.8π θ = 1.π θ = 1.π θ = 1.4π θ = 1.6π θ = 1.8π ime(sec.)

35 Esimaion of he log-ransiion densiy Log Transiion Probabiliy ( θ=π ) x 1 /π x /π

36 y =, y n = π "Perfec" sampling disribuion Sampling disribuion of SMC v/π v/π y =, y n = π "Perfec" sampling disribuion Sampling disribuion of SMC v/π v/π

37 y =, y n = 4π "Perfec" sampling disribuion Sampling disribuion of SMC v/π v/π y =, y n = 5π "Perfec" sampling disribuion Sampling disribuion of SMC v/π v/π

38 Quesions Wha are he principles of designing he inermediae disribuions or he equivalen resampling scheme? How do we know one is beer han anoher? Trade-off beween beer inermediae disribuions and complexiy Raionalize some of he exising resampling schemes 38

39 .1.4 Inference Inference: Ê π h(x ) = m j=1 w(j) h(x (j) ) m j=1 w(j) Esimaion should be done before a resampling sep Rao-Blackwellizaion: For example, if w +1 does no depend on x +1, hen Ê π+1 h(x +1 ) = m j=1 w(j) ) +1 E π +1 (h(x +1 ) x (j) m j=1 w(j) +1 Delayed esimaion (i.e. E π h(x k ) a ime ) is usually more accurae since he esimaion is based on more informaion. Frequen resampling may have adverse effec. 39

40 Resampling Schedule S w π π +1( x ) ( x ) ~ w + 1 RB Es. ~ + 1( x + 1 ) w π x + 1 g ( x ) x Resampling? No Yes S +1 w +1 + S ~ w + 1 g +1 (x +1 x ) w a +1 1 a + 1 π + 1 x π ( x ) ( ) Es. A discree approx.of prob. dis. a ime. A discree approx.of prob. dis. a ime +1. 4/9/3 Paricle Filer and SMC 8