Likelihood ratios and Bayesian inference for Poisson channels

a r X i v :0709.1211v 2 [c s .I T ] 26 N o v 20071Likelihood ratios and Bayesian inference forPoisson channelsAnthony R´e veillac Laboratoire de Math´e matiques Universit´e de La Rochelle Avenue Michel Cr´e peau 17042La Rochelle CedexFranceAbstract —In recent years,infinite-dimensional methods have been introduced for the Gaussian channels estimation.The aim of this note is to study the application of similar methods to Poisson channels.In particular we compute the Bayesian estimator of a Poisson channel using the likelihood ratio and the discrete Malliavin gradient.This algorithm is suitable for numerical implementation via the Monte-Carlo scheme.The result is extended to mixed Poisson-Gaussian channels and to other non-Gaussian channels.Index Terms —Poisson process,Bayesian estimation,Malliavin calculus.Mathematics Subject Classification:94A40,60J75,62C10,60H07.I.I NTRODUCTIONRecently in [12],infinite-dimensional methods have been used to derive a new expression of the conditional mean estimator for infinite-dimensional additive Gaussian channels.More precisely the conditional mean estimator is obtained as the Malliavin derivative of the logarithm of the likelihood ratio.This relationship has been recently applied in [8]in order to link the quadratic risk of the conditional mean estimator to the Monge-Kantorovitch measure transportation theory.Let us give some details about these results.In the general framework of additive Gaussian channel ,an observed signal Y is decomposed into the sum of an input signal X plus an independent Gaussian noise W as Y =ρX +W,(I.1)where ρis the “signal to noise ratio”.In thiscontext the signals “lie”in an abstract Wiener space (W,H,µW )(in particular the input X is an H -valued random variable).This setting contains the case of an observed continuous-time stochastic process (Y t )t ∈[0,T ]related to an input stochastic process (X t )t ∈[0,T ](with values into the Hilbert space H :=L 2([0,T ]))by the following stochastic differential equation,dY t =√ρ∇log l (Y ),(I.3)where Y denotes the sigma field generated by Y ,∇denotes the Malliavin gradient which is a infinite-dimensional counterpart of the usual derivative on R n and l is the likelihood ratio associated to model (I.1)that is,l :=dµYdρ=ρI E X −I E [X |Y ] 2H,where I (X ;Y )denotes the mutual information be-tween X and Y ,defined as,I (X ;Y ):=H ×WlogdµX,Y2Note that (I.2)is an infinite-dimensional counter-part of a well known result for finite-dimensional additive Gaussian channels described as,Y =√ρyx √dµλ0(y )=exp(−αx )αx +λ0dµλ0(y )µX (dx ),y ∈N .(II.1)Now we can state the following lemma which will be extended in Section IV as Proposition IV .6and Corollary IV .8.Lemma II.1.If the Bayesian riskI E |X −I E [X |Y ]|2 is finite thenI E [X |Y ]=λ0m (Y ).(II.2)Proof:Let y in N. m(y+1)−m(y)=αλ0 yµX(dx)(∗) =αdµX(x)µX(dx),P−a.s.=αλ0m(y)I E[X|Y=y]Equality(∗)is justified by a relation of the form iii)of Proposition IV.1.dµ1(y)=e−x(x+1)y,(x,y)∈R+×N, and one obtains,I E[X|Y]=a e−a(1+a)y+b e−b(1+b)ym(Y).(II.3)To obtain results for more general Poisson channels we havefirst to recall some elements of analysis on the Poisson space.III.A NALYSIS ON THE P OISSON SPACEIn this Section we introduce some elements of analysis on the Poisson space in a general frame-work then,we will describe these elements in a concrete example.Let(S,B(S),ν)a measure space whereνis an intensity atomlessσ-finite measure.Define the Pois-son spaceΩS asΩS= y=n k=0δz k,n∈¯N,z k∈S,1≤k≤n , with¯N:=N∪{∞}and let for n k=0δz k, C= n k=0δz k :={z1,...,z n}.(III.1) Define the canonical process(N A)A∈B(S)onΩS asN A(y):=y(A),y∈ΩS.We define theσ-field F S onΩS with F S=σ({y→y(B),B∈B(S)}).There exists a probability measure P S on(ΩS,F S) called the Poisson measure such that,•∀B∈B(S),∀n∈N,P S({y|y(B)=n})=exp(−ν(B))ν(B)nIn this case H[0,T]can be defined in a more tractable way by,H[0,T]= v:[0,T]→R,v(t)= t0˙v s ds,˙v∈L2([0,T]) ,equipped withh1,h2 H[0,T]:= ˙h1,˙h2 L2([0,T]),h1,h2∈H[0,T]. The Malliavin operator∇we introduce will be of interest in Section IV.Let L0(ΩS,F S,P S)be the space of measurable mapping from(ΩS,F S,P S)to R.Definefirst the operator D by,L0(ΩS,F S,P S)→L0(ΩS×S,F S⊗B(S),P S⊗ν)F→D z F(y):=F(y+δz)−F(y). Technical justifications about the measurability of the previous map can be found in[11]and refer-ences therein.This allows us to define the operator ∇.Definition III.1.For F:Ω→R we define∇F as the H S-valued random variable∇A F:= A D z Fν(dz),A∈B(S).Finally in the case of the classical Poisson space the Malliavin derivative∇can be expressed in a different way,for F:Ω[0,T]→R,∇F is a H[0,T]-valued random variable and∇[0,t]F:= t0D s F ds,t∈[0,T].IV.C ONDITIONAL MEAN ESTIMATORS FORP OISSON CHANNELSWe introduce in this section the Bayesian frame-work and we compute the conditional mean es-timator in the setting of Poisson point process. Let X be an input signal with values in a space (H,σ(H))with distributionµX.Consider(Ω,F,P) a probability space and assume the output Y lies in Ω.Until the end of this paper we will denote by Y theσ-field generated by Y.We make the following assumptions,(H1)The“noise”and the“signal”are independent,i.e.the law of the pair(output,input)=(Y,X)is absolutely continuous with respect toP×µX.(H2)For all x in H,µY|X=x(the distribution of Y given X=x)is absolutely continuouswith respect to P and we denote by L thecorresponding Radon-Nikodym density.(H3)L is(σ(H)⊗F)-measurable.(H4)The Bayesian risk I E X−I E[X|Y] 2H with respect toµX isfinite.Then,the following functionH×F→[0,1](x,B)→µY|X=x(B)is a transition probability in the sense of[4,Defini-tion III-2-1p.69].Moreover from[4,PropositionIII-2-1p.69-70]there exists a probability measureµon(H×Ω,σ(H)⊗F)such that,µ(A×B)(IV.1) = AµY|X=x(B)µX(dx),A∈σ(H),B∈F, andµis the joint distribution of(X,Y).Denote by M the marginal distribution ofµon Hdefined by,M(B):=µ(H×B),B∈F.(IV.2) Proposition IV.1is mainly devoted to show the existence of the following transition probabilityΩ×σ(H)→[0,1](y,A)→µX|Y=y(A)(IV.3) and that the couple(M,(µX|Y(·,y))y∈Ω)allows usto recoverµasµ(A×B)= BµX|Y=y(A)M(dy),A∈σ(H),B∈F.(IV.4) Proposition IV.1.If(H1),(H2),and(H3)are satisfied theni)µis absolutely continuous with respect toµX×P and the corresponding Girsanov-Radon-Nikodym density is L.ii)M is absolutely continuous with respect to P with m as Radon-Nikodym density.iii)For M almost all y inΩ,µX|Y=y is absolutely continuous with respect toµX and for y suchthat m(y)=0,the Radon-Nikodym density isgiven by,dµX|Y=ym(y).5iv)For a (σ(H )×F )-measurable function f :H ×Ω→R ,ΩH f (x,y )µX |Y =y (dx )M (dy )= HΩf (x,y )µY |X =x (dy )µX (dx ).Proof:See [2,Theorem 1.8].d P S (y )(IV .6)=exp−αS˙x z ν(dz )×y (S )k =0(1+α˙x (z k )),where y = nk =1δz k .So hypothesis (H2)is satisfied.Finally assume (H4)holds.We recall a result about Bayesian estimator under quadratic loss.Proposition IV .4.The Bayesianestimator(B A )A ∈B (S )isB A (Y )=I E [X (A )|Y ]=H Sx (A )µX |Y (dx ),M −a.e.(IV .7)Remark IV .5.Note that the expression (IV .7)is theoretical and cannot be used in practice.In con-tradistinction,relation (IV .9)obtained below en-ables a numerical approximation of the Bayesian estimator as mentioned in Remark IV .9.In fact it is more tractable to estimate the densities rather than the intensity measures.So we denote by ˙Xthe L 2(S,dν)valued random variable associated to X .For z ∈S (IV .7)can be rewritten as˙B z (y )=I E [˙Xz |Y =y ]=H S˙x z µX |Y (dx,y ),M −a.e.(IV .8)We can state the main result of this Note.It allows us to express the Bayesian estimator of the input as a discrete logarithmic Malliavin gradient of the likelihood ratio m .Proposition IV .6.I E [X A |Y ]=∇A m (Y )6So∇A m(y)= A D z m(y)ν(dz)=α H S L(y,x) A1z/∈C(y)˙x zν(dz)µX(dx) =α H S L(y,x) A˙x zν(dz)µX(dx),asνis atomless,=α H S L(y,x)x(A)µX(dx)=α H S x(A)m(y)µX|Y(dx,y),by iii)of Proposition IV.1.By Proposition IV.4,this leads toI E[X A|Y]=∇A m(Y)Remark IV.7.Neither∇nor D satisfy the chain rule of derivation,and consequently ∇A Fαm(Y),t∈[0,T].(IV.10) Remark IV.9.The nonlinearfilter given by equa-tions(IV.6)and(IV.10)can be numerically approx-imated by evaluating m in(IV.5)by a Monte-Carlo scheme.This computation is really tractable since the Malliavin derivative∇is a difference operator.V.A GENERALIZATION TO A CLASS OFNON-G AUSSIAN CHANNELSIn this Section we give a generalization of results from Section IV.We use some notations and defi-nitions presented in Section VI.Let(M t)t∈[0,T]a normal martingale which satisfy a structure equation of the form(VI.1)and which has the chaos representation property(see Definition VI.4).Let(X t)t∈[0,T]a real-valued input process with X t= t0˙X s ds,t∈[0,T].Assume the output signal(Y t)t∈[0,T]is a normal martingale such thatthe measure of the output given the inputµY|X is absolutely continuous with respect to P with likelihood given byL(y,x)=dµY|X=x2 T0˙x2s1{φs=0}ds× s≤T(1+˙x sφ(s))e−˙x sφ(s).We give a brief explanation of the previous formula (see[6,Theorem36,p.77]).•The continuous martingale parts of Y given Xand Y have the same quadratic variations.•The random measure which define the purejump martingale part of Y given X has inten-sity(1+˙X t)ν(dt)whereνdenotes the randommeasure associated to the pure jump martingalepart of Y.Lemma V.1.With notations of Definition VI.3we have,L(y,x)=∞n=01n!I y n ˙x⊗n1[0,t]nis solution to the stochastic differential equationdZ t=˙x t Z t dy t,t∈[0,T],(V.2) Cf.[5].Furthermore the process defined in(V.1)is also solution of the SDE(V.2)which ends the proof.This formulation of L and the definition VI.5of the Malliavin derivative in this context giveD t L(y,x)=˙x t L(y,x),t∈[0,T].By using the general Bayesian results presented in Section IV we have the following Proposition. Proposition V.2.E[X t|Y]=∇t m(Y)We conclude this section by giving two important examples of normal martingales considered above.•Assume(φt)t∈[0,T]is deterministic.Then(M t)t∈[0,T]has the chaos representationproperty see[1],and(M t)t∈[0,T]can be repre-sented asdM t=i t dB t+φt(dN t−λt dt),M0=0,t∈[0,T] where(B t)t∈[0,T]is a standard Brownian mo-tion,i t=1{φt=0},j t=1{φt=1},and(N t)t∈[0,T]is a Poisson process independent of(B t)t∈[0,T] with intensityνt= t0j sφ2s ds.–Forφt=0,t∈[0,T];then(M t)t∈[0,T]isa standard Brownian motion.•Considerφt=βM t,β∈[−2,0).Then (M t)t∈[0,T]is an Az´e ma martingale.This pro-cess has the chaos decomposition property but its increments are not independent contrary to the previous example.VI.A PPENDIXIn this Appendix we give some further elements of stochastic analysis in the framework of normal martingales.Definition VI.1.A stochastic process(M t)t∈[0,T] defined on a probability space(Ω,F,P)with a right continuousfiltration(F t)t∈[0,T]is a normal martingale in L2(Ω)if it a martingale,that is, I E[M2t]<∞,t∈[0,T]andI E[M t|F s]=M s,0≤s<t≤T,such that,I E[(M t−M s)2|F s]=t−s,0≤s<t≤T.Let(M t)t∈[0,T]a normal martingale on a probabil-ity space(Ω,F,P)with right continuousfiltration(F t)t∈[0,T].Definition VI.2.(M t)t∈[0,T]satisfy a structure equa-tion if there exists an adapted process(φt)t∈[0,T]such that[M,M]t=t+ t0φs dM s,t∈[0,T].(VI.1) Definition VI.3.For n≥1,let L2([0,T])◦n bethe space of symmetric functions f n in n variables. For,f n in L2([0,T])◦n define the iterated stochastic integral I M n(f n)byI M n(f n):=n! T0 t n0... t20f n(t1,···,t n)dM t1...dM t n. For f0in R we let I0(f0):=f0.With the notations of the previous Definition onecan show thatI M n(f n)=n T0I M n−1(f n(∗,t)1[0,t]n−1(∗))dM t,n≥1. Definition VI.4.Denote for n≥1,H n={I M n(f n),f n∈L2([0,T])◦n}.We say that(M t)t∈[0,T]has the chaos representation property ifL2(Ω)=∞n=0H n,that is,for every F in L2(Ω)there exists(f n)n∈N such that f n∈L2([0,T])◦n,n≥1andF=∞n=0I M n(f n).We introduce the Malliavin derivative with respectto(M t)t∈[0,T].Definition VI.5.LetS= n k=0I M k(f k),f k∈L2([0,T])◦k,0≤k≤n,n∈N .We define the Malliavin derivative D as the linear operator from S to L2(Ω×[0,T])byD t I M n(f n)=nI M n−1(f n(∗,t)),d P×dt−a.e.For t in[0,T]define∇t as∇t F= t0D s F ds,F∈Dom(D).R EFERENCES[1]M.´Emery,“On the Az´e ma martingales,”in S´e minaire de Prob-abilit´e s,XXIII,vol.1372of Lecture Notes in Math.,pp.66–87, Springer,Berlin,1990.[2] D.Fourdrinier,“Statistique inf´e rentielle,”Sciences Sup,Dunod,2002.[3] D.Guo,S.Shamai and S.Verdu,“Mutual information andminimum mean-square error in Gaussian channels,”IEEE Trans.Inform.Theory,vol.51,pp.1261–1282,2005.[4]J.Neveu,“Bases math´e matiques du calcul des probabilit´e s,”Masson,1964.[5]N.Privault,“An introduction to stochastic analysis in discreteand continuous settings,”Lecture Notes,2007.[6]P.Protter,“Stochastic integration and differential equations.Anew approach,”vol.21of Applications of Mathematics,Springer-Verlag,Berlin,1990.[7]R.Reiss,“A course on point processes,”Springer Series inStatistics,Springer-Verlag,New York,1993.[8] A.¨Ust¨u nel,“Estimation for the additive Gaussian channel andMongeKantorovitch measure transportation,”Stochastic Process.Appl.,vol.117,pp.1316–1329,2007.[9]S.Verdu,“Poisson Communication Theory,”Invited talk,March251999.The International Technion Communication Day in honor of Israel Bar-David,1999.Available at/verdu/reprints/VerduPoisson1999.pdf [10] E.Wong and M.Zakai,“A characterization of the kernelsassociated with the multiple integral representation of some functionals of the Wiener process,”Systems Control Lett.,vol.2, no.2,pp.94–98,1982.[11]L.Wu,“A new modified logarithmic Sobolev inequality forPoisson point processes and several applications,”Probab.The-ory Related Fields,vol.118,no.3,pp.427–438,2000.[12]M.Zakai,“On mutual information,likelihood-ratios and es-timation error for addaitive Gaussian channel,”IEEE Trans.Inform.Theory,vol.51,no.9,pp.3017–3024,2005.。