Stable approximations of optimal filters

Transcription

1 Sable approximaions of opimal filers Joaquin Miguez Deparmen of Signal Theory & Communicaions, Universidad Carlos III de Madrid. Join work wih Dan Crisan (Imperial College London) and Albero López-Yela (Universidad Carlos III de Madrid). Bayesian Compuaion for High-Dimensional Saisical Models Naional Universiy of Singapore, Sepember 5, 2018

2 Index Inroducion Opimal filering. Sabiliy. Truncaed filers Approximaion of opimal filers: runcaing he likelihood and reshaping he kernel. The normalisaion sequence Bounding he normalisaion consans away from zero (infiniely ofen). Sabiliy of he approximaions Sabiliy of runcaed filers. A meric space of opimal filers. Discussion Summary, an example and some hand-waving.

3 Index Inroducion Opimal filering. Sabiliy. Truncaed filers Approximaion of opimal filers: runcaing he likelihood and reshaping he kernel. The normalisaion sequence Bounding he normalisaion consans away from zero (infiniely ofen). Sabiliy of he approximaions Sabiliy of runcaed filers. A meric space of opimal filers. Discussion Summary, an example and some hand-waving.

4 Inroducion Opimal filering in discree ime Sable filers forgeing iniial condiions Why is sabiliy imporan? Can we approximae unsable filers by sable ones? Are here many sable filers? Work in progress expec some loose ends!!

5 Inroducion Opimal filering in discree ime Sable filers forgeing iniial condiions Why is sabiliy imporan? Can we approximae unsable filers by sable ones? Are here many sable filers? Work in progress expec some loose ends!!

6 Inroducion Opimal filering in discree ime Sable filers forgeing iniial condiions Why is sabiliy imporan? Can we approximae unsable filers by sable ones? Are here many sable filers? Work in progress expec some loose ends!!

7 Inroducion Opimal filering in discree ime Sable filers forgeing iniial condiions Why is sabiliy imporan? Can we approximae unsable filers by sable ones? Are here many sable filers? Work in progress expec some loose ends!!

8 Inroducion Opimal filering in discree ime Sable filers forgeing iniial condiions Why is sabiliy imporan? Can we approximae unsable filers by sable ones? Are here many sable filers? Work in progress expec some loose ends!!

9 Inroducion Opimal filering in discree ime Sable filers forgeing iniial condiions Why is sabiliy imporan? Can we approximae unsable filers by sable ones? Are here many sable filers? Work in progress expec some loose ends!!

10 Sae space Markov models κ κ 2 κ 3 κ X 1 X X 2 X 3 X g 1 g 2 g 3 g Y 1 Y Y Y Random sequences X 0: and Y 1: defined on a probabiliy space (Ω, F, P). Sae-space Markov model X 0 π 0(dx) (prior) X κ (dx x 1) (Markov kernel) Y g (y x )dy (likelihood pdf of observaion) X X, 0, is he sae of he sysem. Y Y, 1, is he observaion. The observaions are condiionally independen given he saes. Noaion: likelihood funcion g y (x) = g (y x)

11 Opimal filering The model: X 0 κ 0, X κ ( x 1), Y g ( x ) The opimal filering problem: recursively compue... he filers π (A) := P (X A Y 1: = y 1:) he predicive measures ξ (A ) := P (X A Y 1: 1 = y 1: 1) Noaion for inegrals: f a es funcion, µ a measure (f, µ) = f (x)µ(dx). Predicion and updae equaions ξ = κ π 1 (f, ξ ) = f (x )κ (dx x 1)π 1(dx 1) π = g y ξ (f, π ) = y (fg, ξ ) (g y, ξ )

12 Sabiliy of he opimal filer Le P(X ) denoe he se of probabiliy measures over X. The sae space Markov model can be represened by a prior disribuion π 0 and an operaor on measures Φ : P(X ) P(X ), defined as Φ (µ) := g y y (fg, κ µ) κ µ (f, Φ (µ)) := (g y, κ µ) If we denoe Φ k (µ) = (Φ Φ 1 Φ k+1 )(µ) hen π = Φ (π 1), π = Φ k (π k ), and, in paricular, π = Φ 0 (π 0). We refer o Φ as a predicion-updae (PU) operaor.

13 Sabiliy of he opimal filer Definiion (Sable filer) The sequence of opimal filers generaed by he PU operaors {Φ } 1 is sable iff lim (f, Φ 0 (µ)) (f, Φ 0 (ν)) = 0 for any pair of iniial measures µ, ν P(X ). We say ha he sequence of opimal filers π = Φ (π 1) is sable, or ha he operaor Φ 0 is sable or simply ha he opimal filer is sable. Various more deailed forms of sabiliy can be defined (P-a.s., for µ << ν, wih specific raes, ec.)

14 Sabiliy of he opimal filer The analysis of filer sabiliy is no an easy ask... Some (relaively) recen references: (Chigansky & Lipser, 2006) sabiliy for a resriced class of es funcions (defined in erms of g and κ ); maringale convergence. (Klepsyna & Vereennikov, 2008; Douc e al., 2009; Gerber & Whieley, 2017) mixing sae process; assumpions no sraighforward o check. (Van Handel, 2009) coninuous-ime filers; links sabiliy wih observabiliy. (Heine & Crisan, 2008) simple condiions (ails of noise disribuions); resriced class of sae space models. To his day, no aemp o assess wheher sable filers are many or few for a given class of models. We adop (and adap) condiions from (Del Moral & Guionne, 2001) and (Del Moral, 2004): sufficien mixing, can be esed on he kernel κ.

15 Index Inroducion Opimal filering. Sabiliy. Truncaed filers Approximaion of opimal filers: runcaing he likelihood and reshaping he kernel. The normalisaion sequence Bounding he normalisaion consans away from zero (infiniely ofen). Sabiliy of he approximaions Sabiliy of runcaed filers. A meric space of opimal filers. Discussion Summary, an example and some hand-waving.

16 Approximae SSM s = Approximae filers We describe a sae space model (SSM) as S = (π 0, κ, g y ) = (π 0, Φ ) and i generaes he sequence of filers π = Φ 0 (π 0). We aim a consrucing modified models, say in such a way ha S c = (π 0, κ c y,c, g ) = (π 0, Φ c ) π c = Φ c 0(π 0) and (f, π ) (f, π c ) < ɛ uniformly over ime. We keep he same π 0 for he wo models. Why??

17 Truncaion scheme 1. Choose a sequence of compac subses c = {C X } Consruc a sequence of runcaed likelihoods { y,c g y g (x) := 1 C (x)g y (x), if x C (x) = 0, oherwise 3. Consruc a sequence of reshaped kernels 4. Le S c := (π 0, κ c, g κ c 1(dx 1 x 0) := κ 1(dx 1 x 0) ρ (dx ) := κ (dx x 1)π 1(dx 1) X \C 1 κ c (dx x 1) := π 1(C 1)κ (dx x ) + ρ (dx ). y,c ) = (π 0, Φ c ), where Φ c (µ) := g y,c κ c µ and π c = Φ c 0(π 0).

18 A picure of he reshaped kernel κ c C κ +1 (dx x ) C +1 C κ c +1 (dx x ) C +1

19 Lemma (Error rese) An approximaion heorem Le S = (π 0, κ, g ) be a sae space model and le c = {C } 1 be a sequence of compac subses of X. The runcaed sae space model S c = (π 0, κ c y,c, g ) yields sequences of predicive and filering probabiliy measures (ξ c and π c, respecively) such ha, for every f B(X ) and every 1, (1 C f, ξ ) = (1 C f, ξ c ), and (1 C f, π ) = (f, π c )π (C ). Theorem (Uniform approximaion) Le S = (π 0, κ, g ) be a sae space model and le c = {C } 1 be a sequence of compac subses of X. Assume every C saisfies (1 C, π ) < ɛ for some 2 prescribed ɛ (0, 1). Then, he runcaed model S c = (π 0, κ c y,c, g ) yields a sequence of filers π c = Φ c 0(π 0) such ha, for every f B(X ), sup (f, π ) (f, π c ) < f ɛ. 1

20 Sabiliy lemma The approximae model S c is useful only if he approximae filers π c = Φ c 0(π 0) are sable... Lemma (Adaped from (Del Moral & Guionne, 2001)) Le c = {C X } >0 be a sequence of compac subses of he sae space X and le Φ c be he runcaed PU operaor induced by he pair (κ, g c ). If he Markov kernels κ have posiive probabiliy densiies k wih respec o a reference measure λ, k ( x 1) = dκ( x 1), dλ and =1 hen he PU operaor Φ c is sable, i.e., inf (x 1,x ) C 1 C k (x x 1) sup (x 1,x ) C 1 C k (x x 1) =, lim (f0, Φc 0(µ)) (f 0, Φ c 0(µ )) = 0 for every µ, µ P(X ) and every f B(X ).

21 Index Inroducion Opimal filering. Sabiliy. Truncaed filers Approximaion of opimal filers: runcaing he likelihood and reshaping he kernel. The normalisaion sequence Bounding he normalisaion consans away from zero (infiniely ofen). Sabiliy of he approximaions Sabiliy of runcaed filers. A meric space of opimal filers. Discussion Summary, an example and some hand-waving.

22 Random observaions & random measures Assume he sequence of observaions Y 1: is random. The sequences of opimal filers & predicive measures are hence random, ξ Y 1: 1 π Y 1: := κ π Y 1: 1 1 (f, ξ Y 1: 1 := g Y ξ Y 1: 1 1 (f, π Y 1: ) = ) = ((f, κ ), π Y 1: 1 ) Y (fg, ξ Y 1: 1 ) (g Y, ξ Y 1: 1 ) Define he σ-algebra generaed by Y 1:, denoed G = σ (Y 1:): The G -measurable r.v. ξ Y 1: 1 is G 1-measurable, π Y 1: is G -measurable. (g Y, ξ Y 1: 1 ) is he normalisaion consan of he (random) filer π Y 1:. The Y [0, ) funcion p (y) := (g Y =y, ξ Y 1: 1 ) is he (predicive) pdf of he random observaion Y condiional on he σ-algebra G 1.

23 Why is he normalisaion sequence imporan? Le c = {C } 1 be a sequence of compacs for runcaion. The approximaion heorem works when π Y 1: ( C ) < ɛ 2, where C = X \C. This is a conrol on he ails of he filer π. If we decompose his facor, π Y 1: ( C ) = (1 C g Y, ξ Y 1: 1 ) (g Y, ξ Y 1: 1 ) runcaion can help wih he numeraor, e.g., (1 C g Y, ξ Y 1: 1 ) sup C...bu no wih he denominaor. In general, ( ) Y P inf (g, ξ Y 1: 1 ) = 0 1 hence π Y 1: ( C ) is hard o upper bound., g Y (x) > 0

24 Why is he normalisaion sequence imporan? Le c = {C } 1 be a sequence of compacs for runcaion. The approximaion heorem works when π Y 1: ( C ) < ɛ 2, where C = X \C. This is a conrol on he ails of he filer π. If we decompose his facor, π Y 1: ( C ) = (1 C g Y, ξ Y 1: 1 ) (g Y, ξ Y 1: 1 ) runcaion can help wih he numeraor, e.g., (1 C g Y, ξ Y 1: 1 ) sup C...bu no wih he denominaor. In general, ( ) Y P inf (g, ξ Y 1: 1 ) = 0 1 hence π Y 1: ( C ) is hard o upper bound., g Y (x) > 0

25 Why is he normalisaion sequence imporan? Le c = {C } 1 be a sequence of compacs for runcaion. The approximaion heorem works when π Y 1: ( C ) < ɛ 2, where C = X \C. This is a conrol on he ails of he filer π. If we decompose his facor, π Y 1: ( C ) = (1 C g Y, ξ Y 1: 1 ) (g Y, ξ Y 1: 1 ) runcaion can help wih he numeraor, e.g., (1 C g Y, ξ Y 1: 1 ) sup C...bu no wih he denominaor. In general, ( ) Y P inf (g, ξ Y 1: 1 ) = 0 1 hence π Y 1: ( C ) is hard o upper bound., g Y (x) > 0

26 Bounding he normalisaion sequence from below So, we need o bound (g Y, ξ ) away from zero... a leas someimes. Assumpion (Predicabiliy) There is some consan γ > 0 such ha E[(g Y, ξ Y 1: 1 ) G 1] γ for every 1. Theorem (Normalisaion sequence) Choose a sae space model S = (π 0, κ, g Y ). If he predicabiliy assumpion holds and g <, hen P-a.s. here exiss an infinie sequence of posiive inegers { n} n 1 such ha ( ) ( ) g Yn, ξ Y 1:n 1 g Y n+1, ξ Y 1:n > γ for every n 1 2 and for some ɛ 2 > 0. lim T T =1 1 { n} n 1 () ɛ 2 > 0 T

27 Bounding he normalisaion sequence from below So, we need o bound (g Y, ξ ) away from zero... a leas someimes. Assumpion (Predicabiliy) There is some consan γ > 0 such ha E[(g Y, ξ Y 1: 1 ) G 1] γ for every 1. Theorem (Normalisaion sequence) Choose a sae space model S = (π 0, κ, g Y ). If he predicabiliy assumpion holds and g <, hen P-a.s. here exiss an infinie sequence of posiive inegers { n} n 1 such ha ( ) ( ) g Yn, ξ Y 1:n 1 g Y n+1, ξ Y 1:n > γ for every n 1 2 and for some ɛ 2 > 0. lim T T =1 1 { n} n 1 () ɛ 2 > 0 T

28 When are he observaions predicable? When are he observaions predicable? Essenially, when he predicive pdf does no vanish wih ime, lim sup p (y) > 0 y Y + some coninuiy. Alernaive sufficien condiions can be saed for he predicabiliy assumpion. A simple example: for a linear-gaussian sysem X 0 N (x 0; m 0, σ 2 0), X N (x ; ax 1, σ 2 u), Y N (y ; bx, σ 2 v ), i is sufficien ha lim sup Var(Y G 1) <.

29 When are he observaions predicable? When are he observaions predicable? Essenially, when he predicive pdf does no vanish wih ime, lim sup p (y) > 0 y Y + some coninuiy. Alernaive sufficien condiions can be saed for he predicabiliy assumpion. A simple example: for a linear-gaussian sysem X 0 N (x 0; m 0, σ 2 0), X N (x ; ax 1, σ 2 u), Y N (y ; bx, σ 2 v ), i is sufficien ha lim sup Var(Y G 1) <.

30 When are he observaions predicable? A more elaborae one... Proposiion Assume ha λ is he Lebesgue measure and he hree condiions below are saisfied: (c1) The condiional pdf g (y x) is Lipschiz w.r.. boh x and y; specifically, here exis consans L 1, L 2 < independen of such ha, for all 1, g y (x) g y (x) L 1 y y and g y (x) g y (x ) L 2 x x. (c2) There exiss m > 0 independen of such ha, for every x X, sup y Y g y (x) m. (c3) There exiss ɛ 0 > 0, arbirarily small, such ha for some consan C dx > 0. inf max 1 x X ξy 1: dx +1 (B(x, ɛ 0)) C dx ɛ0 P-a.s. Then, here exiss γ > 0 such ha E[(g Y, ξ Y 1: 1 ) G 1] > γ.

31 Index Inroducion Opimal filering. Sabiliy. Truncaed filers Approximaion of opimal filers: runcaing he likelihood and reshaping he kernel. The normalisaion sequence Bounding he normalisaion consans away from zero (infiniely ofen). Sabiliy of he approximaions Sabiliy of runcaed filers. A meric space of opimal filers. Discussion Summary, an example and some hand-waving.

32 Sae space model Prior: π 0 Markov kernel: κ has a pdf k, κ (dx x 1) = k (x x 1)λ(dx ), and here is a funcion a : X X such ha a (x 1) = x κ (dx x 1). Likelihood: here is some funcion b : X Y such ha b (x ) = y g y (x )dy. Essenially, an addiive model.

33 Model assumpions Assumpion (MLip) Funcions a : X X, = 1, 2,..., are uniformly Lipschiz, i.e., here exiss a consan L a < such ha a (x) a (x ) < L a x x for every pair (x, x ) X 2 and every > 0. Assumpion (M-) For every 1 here exiss and inverible decreasing funcion s : [0, ) [0, ) such ha and lim r s (r) = 0. k (x x ) s ( x a (x ) ) for every pair (x, x ) X 2, Assumpion (M+) The pdf s k are uniformly upper bounded. Specifically, here exiss a consan C u < such ha sup 1;(x,x ) X 2 k (x x ) < C u < and, lim x a (x ) k (x x ) = 0.

34 The sequence of compac ses Given a consan M <, choose a sequence of closed balls C M := B(l, Mr ) = {x X : x l Mr }, 1 where l X and r [0, ). The cenres saisfy l a (l 1) < MLr, = 1, 2,..., for some consan L <......and he radius {r } 1 is sricly increasing and saisfies ( 1 ) lim s 1 r =, for he funcions s in assumpion (M-). Inuiively, {r } 1 increases a a sufficienly slow rae. Lemma For any given consan M <, if assumpions (MLip) and (M-) hold hen here exiss M < such ha inf (x,x ) C M C M 1 k (x x ) > 1 for every > M.

35 Finie horizon approximaion Le S = (π 0, κ, g y ) be a arge SSM saisfying assumpions (MLip), (M-) and (M+). Using he sequence c = {C M } 1, we consruc he model S c,m = (π 0, κ c,m y,c,m y,c,m, g ), where g = 1 C M g y and { κ c,m κ c,m, for = 1,..., T (reshaped) = κ, for > T, for some finie horizon T <. The PU operaor is Φ c,m and π c,m = Φ c,m 0 (π0). Theorem (Sable approximaion, finie horizon) Le assumpions (MLip), (M-) and (M+) hold. Then, for every pair of consans ɛ > 0 and T <, here exiss M ɛ,t < such ha (f, π ) (f, π c,m ) < ɛ f, 1 T, for every f B(X ) and every M M ɛ,t. Moreover, he PU operaor Φ c,m sable. is

36 A opological inerpreaion Le S denoe he se of sae space models S = (π 0, κ, g y ) ha saisfy assumpions (MLip), (M-) and (M+). We consruc meric spaces (S, D q), q > 1, where and D q(s, S ) := D v (π, π ) := =0 1 q Dv (π, π ) sup π(a) π (A) 1 A B(X ) is he oal variaion disance beween π, π P(X ). Sable runcaed approximaions are dense in he space (S, D q)... Theorem (Sable approximaions are dense) The subse S 0 := {S S : Φ is sable} is dense in he meric space (S, D q).

37 Uniform & sable approximaions Back o random observaions & measures! Assumpion (G) For any ɛ > 0, arbirarily small, here is some ɛ and some M ɛ such ha for every ɛ and every M M ɛ we can find a sequence l Y 1: X saisfying he inequaliy sup x X \B(l Y 1:,Mr ) g Y (x) < ɛ P-a.s. Theorem (Uniform & sable approximaion) Le S = (π 0, κ, g Y ) be a sae space model saisfying assumpions (MLip), (M-), (M+) and (G). If he predicabiliy assumpion also holds, hen for every ɛ > 0 i is possible o choose a sequence of compac ses c = {C } 0 and a runcaed sae space model S c = (π 0, κ c Y,c, g ) such ha he associaed PU operaor Φ c,y is P-a.s. sable and (f, π ) (f, π Y 1:,c ) ɛ f P-a.s., for every 1 and any f B(X ).

38 Skech of he proof: uniform & sable approx. 1. Under he predicabiliy assumpion, P-a.s. here exiss an infinie sequence { n} n 1 such ha ( ) (, ξ Y 1:n 1 g Y n+1 g Yn ), ξ Y 1:n > γ 2. (1) 2. We consruc he sequence of compacs c = {C } { B(l Y 1:, M r ), if { n, n + 1} C = n 1, B(l Y 1:, R ), if / { n} n 1, wih M jus sufficienly large (via assumpion (G)) while R is as large as needed o ensure ha π Y 1: (X \C ) < ɛ Using assumpion (G) again, ogeher wih inequaliy (1), ( ) g Yn,M r n ) n, ξ Y 1:n 1 n ( ) π Y 1:n n X \B(l Y 1:n n, M r n ) = 1 X \B(l Y 1:n n (g Yn n, ξ Y 1:n 1 n ) wih ɛ = 4 ɛ γ. 4. Now he uniform approximaion Theorem yields < 2 ɛ γ = ɛ 2, sup (f, π ) (f, π Y 1:,c ) ɛ f P-a.s., for every f B(X ). 1

39 Skech of he proof: uniform & sable approx. 5. As for he sabiliy of Φ c,y, we noe ha inf (x,x ) C n+1 C n k c n+1 n (x x ) π Y 1:n n (C n ) inf (x,x ) C n+1 C n k n+1 n (x x ) > 1 ɛ 2 n for n > M (and M < ). 6. Therefore, inf k c +1(x x ) > (x,x ) C +1 C 1 > n: n> M n: n> M π Y 1:n n (C n ) inf (x,x ) C n+1 C n k n+1 n (x x ) 1 ɛ 2 n. 7. The laer inequaliy yields sabiliy of Φ c (via he sabiliy Lemma).

40 Index Inroducion Opimal filering. Sabiliy. Truncaed filers Approximaion of opimal filers: runcaing he likelihood and reshaping he kernel. The normalisaion sequence Bounding he normalisaion consans away from zero (infiniely ofen). Sabiliy of he approximaions Sabiliy of runcaed filers. A meric space of opimal filers. Discussion Summary, an example and some hand-waving.

41 Summary An (ongoing) effor o consruc sable approximaions of opimal filers for as general families as possible. Main resuls up o his ime: A uniform approximaion heorem for runcaed filers. A probabilisic characerisaion of he normalisaion consans of opimal filers. A class of meric spaces of opimal filers where sable filers are provably dense. A (somewha resricive) class of paricle filers which admi sable & uniform (over ime) approximaions. Curren version of he manuscrip: D. Crisan, J. M., Sable approximaion schemes for opimal filers, arxiv: [sa.co], hps://arxiv.org/abs/

42 PS. An example & hand-waving We are seeking examples of models which yield unsable filers bu can be approximaed (for a fixed prior π 0) by sable runcaed filers. No so easy: proving eiher sabiliy or insabiliy is usually hard. Good candidaes in he following class of sae space models: The sae space can be pariioned as X = m i=1 X i. The Markov kernel does no ransfer probabiliy mass from X i o X j : κ (A i x j ) = 0 whenever A i X i and x j X j, i j. The resriced likelihoods g y i, (x) = 1X i assumpion (G). A simple one (which can be worked ou): X 0 π 0, y (x)g (x) separaely saisfy X = ax 1 + σx U, U N (0, 1), X Y = X + σ V, V N (0, 1),

43 References Chigansky, P., & Lipser, R On a role of predicor in he filering sabiliy. Elecronic Communicaions in Probabiliy, 11, Del Moral, P Feynman-Kac Formulae: Genealogical and Ineracing Paricle Sysems wih Applicaions. Springer. Del Moral, P., & Guionne, A On he sabiliy of ineracing processes wih applicaions o filering and geneic algorihms. Annales de l Insiu Henri Poincaré (B) Probabiliy and Saisics, 37(2), Douc, R., For, G., Moulines, E., & Prioure, P Forgeing he iniial disribuion for hidden Markov models. Sochasic processes and heir applicaions, 119(4), Gerber, M., & Whieley, N Sabiliy wih respec o iniial condiions in V -norm for nonlinear filers wih ergodic observaions. Journal of Applied Probabiliy, 54(1), Heine, K., & Crisan, D Uniform approximaions of discree-ime filers. Advances in Applied Probabiliy, 40(4), Klepsyna, M. L., & Vereennikov, A. Yu On discree ime ergodic filers wih wrong iniial daa. Probabiliy Theory and Relaed Fields, 141(3-4), Van Handel, R Uniform observabiliy of hidden Markov models and filer sabiliy for unsable signals. The Annals of Applied Probabiliy, 19(3),