c 2003 Society for Industrial and Applied Mathematics

Transcription

1 SIAM J. CONTROL OPTIM. Vol. 42, No. 5, pp c 23 Sociey for Indurial and Applied Mahemaic A VARIATIONAL APPROACH TO NONLINEAR ESTIMATION SANJOY K. MITTER AND NIGEL J. NEWTON Abrac. We conider eimaion problem, in which he eimand,, and obervaion, Y, ake value in meaurable pace. Regular condiional verion of he forward and invere Baye formula are hown o have dual variaional characerizaion involving he minimizaion of apparen informaion and he maximizaion of compaible informaion. Thee boh have naural informaionheoreic inerpreaion, according o which Baye formula and i invere are opimal informaion proceor. The variaional characerizaion of he forward formula ha he ame form a ha of Gibb meaure in aiical mechanic. The pecial cae in which and Y are diffuion procee governed by ochaic differenial equaion i examined in deail. The minimizaion of apparen informaion can hen be formulaed a a ochaic opimal conrol problem, wih co ha i quadraic in boh he conrol and obervaion fi. The dual problem can be formulaed in erm of infinie-dimenional deerminiic opimal conrol. Local verion of he variaional characerizaion are developed which quanify informaion flow in he eimaor. In hi conex, he informaion conerving propery of Bayeian eimaor coincide wih he Davi Varaiya maringale ochaic dynamic programming principle. Key word. Bayeian inference, informaion heory, Legendre-ype ranform, nonlinear filering, ochaic opimal conrol AMS ubjec claificaion. 93E11, 93E2, 94A15, 62F15, 6E1, 6G35 DOI /S Inroducion. Thi aricle inveigae a variaional formulaion of Bayeian eimaion wih a naural informaion-heoreic inerpreaion. The wo direcion of an abrac Baye formula likelihood funcion o poerior diribuion and vicevera are given variaional repreenaion. The forward repreenaion involve he minimizaion of apparen informaion of probabiliy meaure on he pace of he eimand. Thi apparen informaion i made up of wo par:he informaion gain of he meaure over he prior diribuion for he eimand and a reidual erm repreening he informaion value of he obervaion, complemenary o hi. The apparen informaion of probabiliy meaure i greaer han or equal o he oal informaion in he obervaion, wih equaliy if and only if he meaure i he poerior diribuion of he eimand. Thu he forward Baye formula can be hough of a an opimal informaion proceor in ha i balance inpu and oupu informaion. Subopimal proceor appear o have acce o more informaion han here i in he obervaion. The variaional repreenaion of he invere Baye formula involve he maximizaion Received by he edior Augu 2, 21; acceped for publicaion in revied form May 31, 23; publihed elecronically December 17, 23. Thi work wa ared while he econd auhor wa on a period of Sudy Leave viiing LIDS, uppored by EPSRC gran GR/M9739. I coninued during hi vii o he INRIA reearch group Omega and Sigma2. Furher uppor wa provided by he Army Reearch Office under he MURI gran: Daa Fuion in Large Array of Microenor DAAD ; under Cener for Imaging Science ubconrac, gran DAAD John Hopkin Univeriy; he MURI gran: Viion Sraegie and ATR Performance ubconrac Brown Univeriy; and by an INTEL gran. Thi publicaion i alo an oupu from a reearch projec funded by he Cambridge MIT Iniue CMI. CMI i funded in par by he U.K. Governmen. The reearch wa carried ou for CMI by Maachue Iniue of Technology. CMI accep no reponibiliy for any informaion provided or view expreed. hp:// Deparmen of Elecrical Engineering and Compuer Science, and Laboraory for Informaion and Deciion Syem, Maachue Iniue of Technology, Cambridge, MA 2139 mier@mi.edu. Deparmen of Elecronic Syem Engineering, Univeriy of Eex, Wivenhoe Park, Colcheer, CO4 3SQ, UK njn@eex.ac.uk. 1813

2 1814 SANJOY K. MITTER AND NIGEL J. NEWTON of compaible informaion of likelihood funcion on he pace of he eimand. Thi i defined o be he difference beween he informaion in an unpecified obervaion aociaed wih he likelihood funcion and ha par of hi informaion complemenary o he given poerior diribuion. The compaible informaion of likelihood funcion i le han or equal o he informaion gain of he poerior diribuion over he prior, wih equaliy if and only if he likelihood funcion i equivalen o ha provided by he invere Baye formula. Once again, he invere Baye formula can be hough of a an opimal proceor, balancing inpu and oupu informaion. However, in hi cae, raher han appearing o have an addiional ource of informaion, ubopimal proceor loe or fail o make ue of par of he inpu informaion. In ecion 2, he eimand,, and he obervaion, Y, of he Bayeian problem are uppoed o ake value in Borel pace, and Y, Y, repecively. The aring poin i a regular condiional verion of he Baye formula. In ecion 3, he reul are pecialized o he eimaion of diffuion procee wih parial obervaion. In ha conex, he regular condiional probabiliy diribuion can be choen o be coninuou in he obervaion. I alo ha he key propery of being Markovian. Thi mean ha he family of meaure over which apparen informaion i minimized can be rericed o he diribuion of he proce when a finie energy feedback conrol i applied hrough he drif coefficien. Thu, in hi cae, he minimizaion of apparen informaion can be inerpreed in erm of a problem in ochaic opimal conrol. Thi i explored in ecion 4. The dual variaional problem for diffuion procee i developed in ecion 5. One inerpreaion of i i a a problem in infinie-dimenional deerminiic opimal conrol. The opimal rajecory of he dual problem i a likelihood filer for he proce in revered ime, from which he correponding nonlinear filer can be found. Thi give a new inerpreaion o a connecion beween an opimal conrol problem in one ime direcion and a nonlinear filer in he oher which wa made for nondegenerae diffuion in [6] via he Hopf ranformaion and ued o give exience and uniquene reul for he unnormalized condiional deniy equaion wih unbounded obervaion. The reul of ecion 3 5 are eablihed under fairly weak condiion. In paricular, hey include he cae of degenerae diffuion. In he conex of eimaor for diffuion procee, here i a local verion of he variaional formulaion which characerize flow rae of informaion and how ha Bayeian proceor are conervaive in he ene ha hey balance inpu and oupu flow rae. Thi i he ubjec of ecion 6. A variaional repreenaion of he Fokker Planck equaion for diffuion procee i dicued in [1]. Thi involve he minimizaion of he energy of drif coefficien over hoe ha give rie o a paricular e of marginal deniie. There, a here, he modificaion of he drif coefficien can be inerpreed a he applicaion of a conrol erm, which re-expree he variaional problem a one in opimal conrol. The wo problem are omewha differen hough. In paricular, he conrol admied in [1] give rie o muually ingular raniion probabiliie, which are cerainly no permied in he preen conex. A preliminary accoun of ome of he reul herein wa repored in [11]. 2. A variaional formulaion of Bayeian eimaion. Le Ω, F, Pbea probabiliy pace,, and Y, Y Borel pace, and :Ω and Y :Ω Y meaurable mapping wih diribuion P, P Y, and P Y on, Y, and Y,

3 A VARIATIONAL APPROACH TO NONLINEAR ESTIMATION 1815 repecively. Suppoe ha H1 here exi a σ-finie reference meaure, λ Y,onY uch ha P Y P λ Y. Thi could be P Y ielf. Le Q : Y [, be a verion of he aociaed Radon Nikodym derivaive, and { } 2.1 Ȳ = y Y :< Qx, yp dx < ; hen Ȳ Y and P Y Ȳ =1. LeH : Y, + ] be defined by 2.2 Hx, y = logqx, y if y Ȳ, oherwie; hen P Y : Y [, 1], defined by 2.3 P Y A, y = A exp Hx, yp dx exp Hx, yp dx, i a regular condiional probabiliy diribuion for given Y ; i.e., P Y,y i a probabiliy meaure on for each y, P Y A, iy-meaurable for each A, and P Y A, Y =P A Y a.. Equaion coniue an oucome-by-oucome abrac Baye formula, yielding a poerior probabiliy diribuion for for each oucome of Y. Of coure, for any y belonging o a e of P Y -meaure zero, P Y,y depend on he choice of verion of he Radon Nikodym derivaive Q. However, in paricular example, we can ofen find a verion uch ha P Y A, i coninuou for each A. Le P be he e of probabiliy meaure on, and H he e of, + ]-valued, meaurable funcion on he ame pace. For P, ˆP P, and H H, we define h P ˆP d = log P d ˆP d P if P ˆP and he inegral exi, oherwie; i H = log exp HdP if < exp HdP <, 2.5 oherwie; H, P = Hd P if he inegral exi, oherwie. I i well known ha he relaive enropy h P ˆP can be inerpreed a he informaion gain of he probabiliy meaure P over ˆP. In fac, any verion of logd P /d ˆP i a generalizaion of he Shannon informaion for. For almo all x, i i a meaure of he relaive degree of urprie in he oucome = x for he wo diribuion P and ˆP.Thuh P ˆP i he average reducion in he degree of urprie in hi oucome ariing from he accepance of P a he diribuion for, raher han ˆP.

4 1816 SANJOY K. MITTER AND NIGEL J. NEWTON If we inerpre exp H a a likelihood funcion for, aociaed wih ome unpecified obervaion, hen Hx i he reidual degree of urprie in ha obervaion if we already know ha = x, and i H i he oal degree of urprie in ha obervaion, i.e., he informaion in he unpecified obervaion, if all we know abou i i prior P. In wha follow we hall call H he -condiional informaion in he unpecified obervaion and i H he informaion in ha obervaion. Of coure, H, y and, repecively, ih,y are he -condiional informaion and he informaion in he obervaion ha Y = y. Propoiion 2.1. Suppoe ha H1 i aified, and H and P Y are a defined above. Then for any y uch ha 2.7 Hx, y exp Hx, yp dx <, where + exp =, { i ih,y = min h P P + H,y, P } ; P P { ii hp Y,y P = max i H } H,P Y,y ; H H iii P Y,y i he unique minimizer in 2.8; iv if H i a maximizer in 2.9, hen here exi a real conan K uch ha H =H, y+k a.. Proof. Ify Ȳ and 2.7 hold, hen hp Y,y P <, ih,y >, and H,y L 1 P Y,y. Thi i alo rue if y/ Ȳ ince, in ha cae, H,y= and P Y,y=P. Thu i i clear ha he minimum in 2.8 i le han +, and he maximum in 2.9 i greaer han. Suppoe ha, for P P, h P P < and H,y L 1 P. I readily follow ha P P Y,y, o ha h P d P = log P dp Y x, y + log x, y P dx, dp Y dp and 2.1 h P P + H,y, P = ih,y + h P P Y,y. I i eay o how ha, for any P P, he relaive enropy funcional h P i nonnegaive, evaluae o zero a P, and i ricly convex on he ube of P for which i i finie. Thi eablihe par i and iii. Suppoe now ha, for H H, i H > and H L 1 P Y,y. Le P be defined by 2.3 wih H replacing H,y. I readily follow ha P Y,y P, and o i H H d = log P dp dp Y = log, y dp dp Y log d P, y.

5 A VARIATIONAL APPROACH TO NONLINEAR ESTIMATION 1817 Thu 2.11 i H H,P Y,y = hp Y,y P hp Y,y P. Suppoe ha here i a e A for which P Y A, y =bu P A >. Le P be defined by P B = P A C 1 P A C B for all B. Then hp Y,y P <hp Y,y P, and o any maximizer in 2.11 mu be aboluely coninuou wih repec o P Y,y. I i eay o how ha, for any P P, he relaive enropy funcional h P i nonnegaive, evaluae o zero a P, and i ricly convex on he ube of P coniing of meaure ha are aboluely coninuou wih repec o P. Thi eablihe par ii and iv. Remark 1. If he muual informaion beween and Y i finie, dpy 2.12 log dp Y <, dp P Y Y hen here exi a verion of Q for which 2.7 i aified for all y. Remark 2. Propoiion 2.1 i a pecial cae of an energy-enropy dualiy ha play a major role in aiical phyic and in he heory of large deviaion. More general reul of hi naure are widely available in he lieraure. See, for example, [5]. Our aim in hi ecion i o provide an informaion-heoreic inerpreaion of he reul in he Bayeian conex. The imple proof we provide here make ue of he pecial naure of ha conex. Par i and ii of Propoiion 2.1 boh concern he proceing of informaion over and above ha in he prior P. In par i, he ource of addiional informaion i he obervaion ha Y = y. The abrac Baye formula exrac he par of hi informaion perinen o, hp Y,y P, and leave he reidual informaion, H,y,P Y,y. One can hink of he inpu informaion a being held in he likelihood funcion, exp H,y, and he exraced informaion a being held in he diribuion, P Y,y. An arbirary eimaion procedure ha poulae P a a poobervaion diribuion for appear o have acce o addiional informaion, in ha i yield an informaion gain on of h P P, and a reidual informaion of H,y, P. The um of hee wo erm he erm in bracke on he righ-hand ide of 2.8 i ricly greaer han he acual informaion available, ih,y, unle P = P Y,y. We hall call i he apparen informaion of he eimaor P. Implici in he inerpreaion of h P P a an informaion gain i he aumpion ha P repreen a raional belief abou given he prior and ome addiional knowledge, uch a an obervaion. In par ii, he ource of addiional informaion i he poerior diribuion, P Y,y. The aim now i o poulae an obervaion wih likelihood funcion exp H which would give rie o hi diribuion. The inpu informaion here, hp Y,y P, i merged wih he reidual informaion of he poulaed obervaion, H,P Y,y, and he reul i greaer han or equal o he oal informaion in he poulaed obervaion, i H, wih equaliy if and only if he obervaion i compaible wih P Y,y in he ene of par iv of he propoiion. The erm in bracke on he righ-hand ide of 2.9 can be hough of a ha par of he informaion in he poulaed obervaion compaible wih P Y,y. We hall call i he

6 1818 SANJOY K. MITTER AND NIGEL J. NEWTON compaible informaion of he likelihood funcion exp H. Anoher inerpreaion i ha he inpu informaion, hp Y,y P, i proceed o produce compaible informaion reuling in a ne lo of informaion excep when he proceor i opimal. Throughou he re of he paper, he apparen informaion and compaible informaion will be denoed by F P,y and G H,y, i.e., F P,y=h P P + H,y, P, G H,y=i H H,P Y,y. A 2.1 and 2.11 how, he minimizaion of F i equivalen o he minimizaion of he informaion exce of he eimaor P, h P P Y,y, and he maximizaion of G i equivalen o he minimizaion of he informaion defici of he likelihood funcion exp H, hp Y,y P. In fac a wa poined ou by an anonymou referee, hee inerpreaion ill hold in he abence of 2.7. However, in no idenifying he ource informaion or he exraced informaion, hey do no how he informaion proceing apec of Bayeian eimaion in quie he ame way a he quaniie F and G. Moreover, F and G make clear he compromie involved in Bayeian eimaion. Par i of he propoiion how how P Y,y compromie beween being cloe o he prior P and fiing wih he obervaion Y = y, wherea par ii how how H,y or i equivalen compromie beween holding a lo of informaion bu no oo much reidual informaion. Of coure i i poible o give oher variaional characerizaion of P Y,y. For example, one could conider i a he minimizer of he oal variaion norm of he difference meaure P P Y,y. However, uch characerizaion lack he informaion-heoreic inerpreaion dicued above: F and G are naural error meaure for ubopimal eimaion procedure. The characerizaion 2.8 could be ued a a bai for approximaion. For example, we may wih o approximae a poerior diribuion by a dicree law on a finie pariion of. The ize of he pariion may be fixed, bu we may be able o chooe he law and he deail of he pariion by mean of a finie number of parameer. The characerizaion 2.8 could form he bai of an opimizaion wih repec o hi e of parameer. Similarly, he characerizaion 2.9 could be ued a a bai for he udy of modeling error, in ha i how he informaion lo ariing from he ue of an incorrec likelihood funcion. Since he ue of an incorrec prior, P e wih P e P, wih a Bayeian procedure i equivalen o he ue of he incorrec likelihood funcion exp H e,y = exp H,y dp e dp, 2.9, wih H = H e,y, alo how he informaion lo ariing hrough he ue of an incorrec prior. Furhermore, if here were any uncerainy in he likelihood funcion or he prior, he reuling informaion lo could be udied by mean of game-heoreic mehod. Propoiion 2.1 i an inance of a Legendre-ype ranform beween he relaive enropy of probabiliy meaure and he logarihm of he exponenial momen of realvalued random variable. A imilar ranform occur in he characerizaion of Gibb meaure in aiical mechanic [8]. In ha conex,, iheconfiguraion pace of a phyical yem he Careian produc of a number, N, of idenical pace, H i a Hamilonian repreening he energie of he configuraion, and F i he free energy

7 A VARIATIONAL APPROACH TO NONLINEAR ESTIMATION 1819 of he probabiliy meaure P wih repec o he reference meaure, P, and H. A Gibb meaure repreen a hermodynamic ae of he yem in hermodynamic equilibrium. If N i finie, hen here i only one Gibb meaure, and i ake he form 2.3. Gibb heory come ino i full richne only when N i infinie, in which cae here may be muliple Gibb meaure, and formulae uch a 2.3 are no longer appropriae. However, variaional characerizaion are. We noe ha he Bayeian eimaor can be een o compromie beween being cloe o he prior and fiing wih he obervaion in exacly he ame way ha a hermodynamic yem in equilibrium compromie beween maximizing enropy and minimizing average energy. 3. Pah eimaor. The echnique of ecion 2 are pecialized here for he cae in which he eimand,, and obervaion, Y, are, repecively, coninuou R n - and R d -valued procee governed by he following Iô inegral equaion: = + µ, Y = b, d + g d + W for T, σ, dv for T, where,v R n, µ i a law on R n, B n, Y,W R d, and b, σ, and g are meaurable mapping. Under uiable regulariy condiion, hee equaion will be unique in law and have a weak oluion Ω, F, F,P,V,W,, Y, i.e., a filered probabiliy pace upporing an n + d-dimenional Brownian moion V,W and an n + d-dimenional emimaringale, Y uch ha 3.1 and 3.2 are aified for all. The abrac pace, and Y, Y of ecion 2 now become he pace C[,T]; R n, B T and C[,T]; R d, B T of coninuou funcion, opologized by he uniform norm. We coninue o ue he noaion, and Y, Y, hough, for he ake of breviy. Le λ Y be Wiener meaure on Y, Y. Under uiable condiion on µ, b, σ, and g, we migh expec H1 o be aified and he muual informaion, E logdp Y /dp λ Y, Y, o be finie. Thi will allow u o proceed a in ecion 2 o conruc a funcion H on Y, and a correponding regular condiional probabiliy, P Y, uch ha 2.7 hold for all y. Furhermore, if we can how ha P Y,y P, hen we hall be able o conruc a coninuou ricly poiive maringale M y on Ω uch ha dp Y,y M y, = E dp F for T, where F i he filraion generaed by he proce. I will hen follow from he Cameron Marin Giranov heory ha M y, = M y, exp U y, d b, d σ, U y, 2 d 2 for ome progreively meaurable R n -valued proce U y. P Y,y will hen be he diribuion of a conrolled proce, y, aifying an equaion like 3.1, bu wih a differen iniial law and wih a conrol erm, σσ,u y,, enering he drif coefficien. The ue of he progreively meaurable conrol Ũ inead of U y will reul in a proce having a diribuion whoe apparen informaion relaive o

8 182 SANJOY K. MITTER AND NIGEL J. NEWTON P,H,y i greaer han or equal o ha of y. Thu, a lea in par, he variaional characerizaion of ecion 2 will become a problem in ochaic opimal conrol. We migh alo expec P Y,y o be Markov a lea for almo all y, in which cae i will be appropriae o reric admiible conrol Ũ o feedback conrol of he form u,. I hould alo hen be poible o define regular condiional raniion probabiliie for P Y. Wih hi in mind, le χ, T be he coordinae proce on, and 3.4 = σχ r, r for T. We hould be able o conruc regular condiional probabiliie uch ha, for all A T, 3.5 P Y A, y = P + Y : T R n C[, T ]; R d [, 1] R n P + Y A, z, y y, T P Y χ 1 dz,y. Thee will have variaional characerizaion in erm of he correponding regular condiional probabiliie for he prior, P, and appropriaely conruced likelihood funcion. Thi will lead oward a localized verion of he reul of ecion 2. In wha follow, we develop he above idea in a rigorou manner. We do hi by placing conrain on b and σ uch ha 3.1 ha a rong oluion and hen ue he echnique of ochaic flow. Thi ha he advanage ha we are able o include problem wih degenerae diffuion coefficien, which are imporan in many area of applicaion. In fac our approach alo applie o ome problem no aifying a hypoellipiciy condiion. The conrain we place on µ, b, σ, and g alo fi well wih Clark robune idea ee [2]. Thee lead o an explici funcion H and correponding regular condiional probabiliy, P Y, ha i Markov for every y. They alo admi unbounded obervaion funcion g, which are needed in he linear cae. We uppoe ha µ, b, σ, and g aify he following echnical condiion: H2 here exi an ɛ> uch ha exp ɛ z 2 µdz < ; R n H3 σ i bounded, and b and σ are uniformly Lipchiz coninuou on compac e and differeniable wih repec o he componen of z, he derivaive being coninuou and bounded; H4 g ha coninuou fir, econd, and hird derivaive, and here exi C < and α< uch ha for all z R n g z z i i C, 2 g z z i,j i z j C1 + z, and 3 g z z i z j z k C1 + z α. i,j,k

9 A VARIATIONAL APPROACH TO NONLINEAR ESTIMATION 1821 I follow from H3 ha 3.1 ha a rong oluion Φ : R n, o ha on he probabiliy pace Ω, F, F,P,, V,W upporing an R n -valued random variable wih diribuion µ, and n + d-dimenional vecor Brownian moion V,W, independen of, =Φ,V, F ; T i a coninuou emimaringale aifying 3.1. See, for example, [15]. I follow from H2 H4 ha E T g 2 d <, and from hi and he independence of and W i follow by andard reul ee, for example, [9] ha H1 i aified when he reference meaure λ Y i he Wiener meaure and he Radon Nikodym derivaive ake he form dp T Y, Y = exp g dy 1 T 3.6 g 2 d. dp λ Y 2 In order o develop he repreenaion of Propoiion 2.1, we fir need a verion of hi ha i well defined for all y. Under H2 H4 he proce g, F, T i a emimaringale, and o i i poible o inegrae by par in 3.6 and define Q a any meaurable funcion uch ha, for each y, T Q, y = exp y T g T y dg 1 T 3.7 g 2 d. 2 See [2] and [3]. I can alo be hown ee, for example, [13], [14] ha he reuling regular condiional probabiliy, P Y, i coninuou in y in he ene of he opology aociaed wih he convergence of mean of bounded meaurable funcion, ha 3.8 < EQ, y < for all y, and ha 3.9 EQ, y logq, y EQ, y 2 <. Thu he e Ȳ of 2.1 can be aken o be he enire pace Y in hi cae, and 2.7 i aified for all y. Propoiion 2.1 can hu be applied for each y, and H = logq. We can now pli he pah eimaion problem a uggeed by 3.5. For any z R n and any T, le z, ; T be he oluion of 3.1 on he inerval T wih iniial condiion z, = z, and le be a meaurable funcion uch ha H p :[,T] [,T] R n Y R 3.1 H p,, z, z,,y= y g z, +y gz y r dg z, r g z, r 2 dr for T. The fac ha uch a funcion exi follow from he rong oluion hypohei H3, a doe he decompoiion 3.11 H, y =H p,,,,y+h p, T,,, T,y.

10 1822 SANJOY K. MITTER AND NIGEL J. NEWTON H p,, z,, i he equivalen of H for he problem of eimaing he pah r z,, r given he obervaion Yr z,, r, where Y z, = g z, r dr + W W for T. In paricular, H p, T, z,, i he equivalen of H for he problem of eimaing z, given Y z,. Le vz,, y be he minimum apparen informaion for hi problem; hen, according o Propoiion 2.1 i, 3.12 vz,, y = log E exp H p, T, z, z,,y. I now follow ha, for any A, P Y A, y = E1 A exp H p,,,,y v,,y 3.13, E exp H p,,,,y v,,y and from Jenen inequaliy and 3.9 i follow ha H p,,χ,,y+vχ,,y aifie 2.7 for all. So, from Propoiion 2.1, he pah meaure P Y rericed o i he unique probabiliy meaure on ha minimize he apparen informaion 3.14 F P,,y=h P, P, + H p,,χ,,y, P, + vχ,,y, P,, where P, i he rericion of P o. I alo eaily follow ha he minimum apparen informaion in 3.14 doe no depend on. Thee argumen how ha he variaional form of he pah eimaion problem 3.1, 3.2 can be inerpreed in erm of dynamic programming, wih value funcion v. For each we can pli he problem ino wo ubproblem:he eimaion of z, for each z reuling in a minimum apparen informaion of vz,, y, followed by he eimaion of,, where v,,y play a par in he likelihood funcion. v,,y ummarize ha par of he likelihood funcion aociaed wih incremen of Y afer ime. The fir ubproblem can be inerpreed in erm of ochaic opimal conrol, where he co i he apparen informaion of he conrolled proce. Thi i developed in he nex ecion. 4. A ochaic conrol formulaion. We conider he fir variaional ubproblem dicued above wih =. In keeping wih he commen above on dynamic programming, i urn ou ha we need conider only feedback conrol. Alo, becaue conrol are inended o produce a change in meaure of he form 3.3, i i appropriae o le he conrol ener he drif hrough he map z az, where a = σσ. Conider he following conrolled equaion: = θ + b,+a,u 4.1, d + σ, dṽ, where he iniial condiion, θ, i nonrandom. Le U be he e of meaurable funcion u : R n [,T] R n wih he following properie: U1 u i coninuou, U2 EΓ u = 1, where 4.2 T Γ u = exp u σ θ,, dv 1 T 2 and Ω, F,P, V, and z, are a defined in ecion 3. σ u θ,, 2 d,

11 A VARIATIONAL APPROACH TO NONLINEAR ESTIMATION 1823 Lemma 4.1. If b and σ aify H3 and u U, hen 4.1 ha a weak oluion and i unique in law. Proof. From H3 and U1 i follow ha T P σ u θ,, 2 d < =1. Thi, ogeher wih U2 and Giranov heorem, how ha V u, defined by 4.3 V u = V σ u θ,, d, i a andard Brownian moion under he probabiliy meaure P u, defined by dp u 4.4 dp =Γu. Thi how ha Ω, F, F,P u, θ,,v u i a weak oluion of 4.1. Nex, uppoe ha Ω, F, F, P,,Ṽ i a weak oluion of 4.1, and, for each naural number N, le τ N : [,T] be defined by τ N x = inf{ : x N} T. Since i coninuou, P τ N T = 1. Alo, ince u aifie U1, 1 τn Ẽ exp σ u 2, 2 d <, and o, from a andard variaion of Novikov heorem ee, for example, Theorem 6.1 in [9], i follow ha M, F, T, where M = exp u σ, dṽ 1 σ u 2 4.5, 2 d i a local maringale wih repec o he equence of opping ime τ N ; N = 1, 2,... Le Ṽ N = Ṽ + τn σ u, d; hen, by Giranov heorem, Ṽ N i a andard Brownian moion under he probabiliy meaure P N, defined by d P N = M τn d P. Le ; T be he filraion on, generaed by he coordinae proce χ. Since τn =Φ τ N θ, Ṽ N for T, where Φ i he rong oluion o 3.1, he law of rericed o τn i idenical o ha of θ, under P u, rericed o he ame igma-field. Finally, for any A, P A, τ N =T = P A P A, τ N <T P A,

12 1824 SANJOY K. MITTER AND NIGEL J. NEWTON and o, ince he even on he lef-hand ide each belong o one of τn ; N =1, 2,..., he law of on i idenical o ha of θ, under P u. Le Ω, F, F, P,,Ṽ be a weak oluion of 4.1 for ome u U. We define he co for conrol in U a he apparen informaion of he reuling diribuion of, P. Thi i meaured relaive o he prior P θ, he diribuion of θ, and H p,t,θ,,y a defined in Ju, θ, y =h P P θ, + H p,t,θ,,y, P = 1 2Ẽ T Ẽ T σ u, 2 d y T gθ+ 1 2Ẽ + oherwie, T g 2 d y T y Lg + Dgau, d if he inegral exi, where L i he differenial operaor aociaed wih, L = i b i a i,j, z i 2 z i z j and D i he row-vecor jacobian operaor, D =[ / z 1 / z 2 / z n ]. The co funcional ha a more appealing form in he pecial cae ha he obervaion pah, y, i everywhere differeniable: Ju, θ, y = 1 T σ u 2Ẽ, 2 + ẏ g 2 d 1 T 4.7 ẏ 2 d. 2 Thi involve an energy erm for he conrol and a lea-quare erm for he obervaion pah fi. Thee correpond o he wo erm in Baye formula repreening he degree of mach wih he prior diribuion and he obervaion pah. The opimal conrol problem 4.1, 4.7 can be hough of a a ype of energy-conrained racking problem. The opimal conrol, under which he diribuion of i he regular condiional probabiliy diribuion P Y,y, i derived in he following heorem. Theorem 4.2. Suppoe ha b, σ, and g aify H3 and H4, and le he funcion u : R n [,T] Y R n be defined by i,j 4.8 u = Dv, where v i a defined in Then, for each y Y, u,,y belong o U, and for all θ R n, y Y, and P P no necearily he diribuion of a conrolled proce, 4.9 Ju,,y,θ,y h P P θ, + H p,t,θ,,y, P. Proof. The proof i in hree par. The fir ue he mehod of ochaic flow o eablih a ochaic repreenaion formula for u, 4.2. The econd prove he aemen of he heorem for nondegenerae yem wih bounded coefficien. Finally, a runcaion argumen i ued o exend hi reul o he general cae. Only he ime-homogeneou cae b and σ no dependen on i reaed in order o avoid exceive noaion. The argumen exend in an obviou way o he general cae.

13 A VARIATIONAL APPROACH TO NONLINEAR ESTIMATION 1825 Sandard momen bounding argumen ee, for example, Theorem 4.6 in [9] how ha for each naural number m here exi a C m <, no depending on z or, uch ha up E z, 2m C m 1+ z 2m 4.1 T 4.11 and up E Ψ z, 2m C m, T where Ψ z, R n n ; T i he oluion of he equaion of fir-order variaion aociaed wih z,, 4.12 Ψ z, = I + Db z, r Ψ z, r dr + i Dσ i r z, Ψ z, r dv i,r. Here and in wha follow, σ i i he ih column of σ, and V i, i he ih componen of V. For any z, z R n and any T z, z, =z z+ b z, r b z, r dr + σ z, r σ z, r dv r, and o for any naural number m here exi a C m <, no depending on,, z, or z, uch ha E up z, r r r z, 2m 3 z 2m 1 z 2m + E up C m z z 2m + +E up r r r E up q r r 2m bq z, bq z, dq 2m σq z, σq z, dv q z, q q z, 2m dr, where we have ued Doob ubmaringale inequaliy, 4.1, H3, and andard bound for he momen of ochaic inegral. I hu follow from he Gronwall lemma ha 4.13 E up z, z, 2m C m expc m T z z 2m for all z, z,. T Similarly, 4.14 E up z, 2m C m 1 + z 2m for all z,, T and o for any ɛ> and any bounded e A R n here exi a C< uch ha P up z, >C <ɛ/4 for all z, A [,T]. T From H3 and H4 i follow ha DLg i uniformly coninuou on compac, and o for any η> here exi a δ> uch ha if z, z A and z z <δ, P DLg z, DLg z, >η, up z, z, C <ɛ/2, up T T

14 1826 SANJOY K. MITTER AND NIGEL J. NEWTON o ha 4.15 P up T DLg z, DLg z, >η <ɛ. The polynomial growh of DLg ogeher wih 4.14 and he Vallée Pouin heorem how ha, for any <p<, he family { } DLg z, p ; z A, T up T i uniformly inegrable. Thi and 4.15 how ha for any <p< E up DLg z, DLg z, 4.16 p = o z z T uniformly on A [,T]. Similar argumen how ha Dg z, and DDgσ i z, for i =1, 2,...,n have he ame propery. I follow from he mean-value heorem ha z, z, =z z+ + i Db α,r z, r Dσ i αi,r z, r, Db z, +1 α,r r z, z, r, Dσ i z,, r z, dr +1 α i,r r z, z, r r z, dv i,r, where < α i,r < 1 and α i,r i F r -meaurable for each i. The above coninuiy properie, Hölder inequaliy, and echnique imilar o hoe ued o prove 4.13 now how ha for any <p< 4.17 and E up T z, z, Ψ z, z z p = o z z p, 4.18 E Θz,, y Θ z,, y ξz,, yθz,, yz z p = o z z p, boh uniformly on A [,T], where and ξz,, y =y T y Dgz+ i + T Θz,, y = exp H p, T, z, z,,y T y T y DLg z, y T y DDgσ i z, T Ψ z, d g z, Ψ z, dv i, Dg z, Thu Dρ = EξΘ, where ρ = EΘ. Now, Jenen inequaliy how ha Ψ z, d and o 4.2 inf ρz,, y inf expe logθz,, y >, z A, T z A, T u z,, y = Eξz,, yθz,, y. EΘz,, y

15 A VARIATIONAL APPROACH TO NONLINEAR ESTIMATION 1827 We now conider he pecial cae in which y i differeniable wih Hölder coninuou derivaive, b and g are bounded, and here exi an ɛ> uch ha 4.21 z az z ɛ z 2 for all z, z R n. In hi cae ρ i coninuouly differeniable wih repec o, i wice coninuouly differeniable wih repec o z, and by a andard exenion of he Feynman Kac formula aifie he following parial differenial equaion ee, for example, [7]: ρ + Lρ + ẏ g gρ = onr n,t, ρ,t,y=1. Since v = logρ, he value funcion v aifie v + Lv 1 2 DvaDv ẏ g g = onr n,t, v,t,y=. Now, becaue of 4.1, 4.11, and he boundedne of g and Dg, u,,y i alo bounded and, by Novikov heorem, aifie U2. We have hu hown ha in hi pecial cae u,,y U. Le V and P be abbreviaion for V u,,y and P u,,y, repecively, where, for u U, V u and P u are a defined by 4.3 and 4.4. Then Iô rule and 4.23 how ha T =v θ, T,T,y=vθ,,y+ ẏ 1 2 g g 1 2 σ u 2 θ,,,y d T u σ θ,,,y dv. A wa poined ou in he proof of Lemma 4.1, Ω, F, F,P, θ,,v iaweak oluion of 4.1, and o, ince g, u,,y and σ are bounded, T vθ,,y=e 1 2 σ u ẏ 1 2 g g θ,,,y d = Ju,,y,θ,y. By definiion, vθ,,y i he minimum apparen informaion, and o we have eablihed 4.9 in hi pecial cae. A conequence of 4.9, and he uniquene of he meaure minimizing apparen informaion, i ha he diribuion of when u = u,,y i he regular condiional diribuion of θ, given ha Y = y. Thu, in hi pecial cae, Γ u,,y = Θθ,,y ρθ,,y a.. Nex, uppoe ha he addiional conrain placed on y, b, g, and σ are removed. For any naural number N, le b N z =bz exp z 2 /N, g N z =gz exp z 2 /N, σ N z = [ σn 1 I ] an n 2n marix,

16 1828 SANJOY K. MITTER AND NIGEL J. NEWTON and le y N be a equence of differeniable elemen of Y wih Hölder coninuou derivaive uch ha y y N. Then b N and g N are bounded and σ N aifie 4.21, b N, σ N, and g N aify H3 and H4 uniformly in N, and b N, σ N, g N, Db N, σ N / z i, and Dg N converge o b, [σ ], g, Db, [ σ/ z i ], and Dg repecively uniformly on compac. We add he ubcrip or upercrip N o, Ψ, Θ, ec. o indicae ha y, b, g, and σ have been replaced by y N, b N, g N, and σ N in he variou definiion and ha V ha been replaced by he 2n-dimenional Brownian moion, V,B. Now z, N,z, = + bn r z, b N r N,z, dr + σ z, r br z, b N r z, dr N 1 B B. σr N,z, dv r Argumen imilar o hoe ued o prove 4.13, 4.17, and 4.18 how ha, for any naural number m and any bounded e A R n, E up z, N,z, m, T E up Ψ z, Ψ N,z, 2m, T 4.25 E Θz,, y ΘN z,, y N 2m, and E ξz,, y ξn z,, y N 2m, all uniformly on A [,T]. Thi, Hölder inequaliy, and 4.19 how ha 4.26 u N,,y N u,,y uniformly on A [,T]. Thu u,,y aifie U1. I follow from 4.24 and 4.26 ha u θ,,,y u N N,θ,,,y N in probabiliy, o ha up T Γ u N,,y N 4.27 N Γ u,,y in probabiliy. I alo follow from 4.25 and 4.19 ha 4.28 Γ u N,,y N N = Θ Nθ,,y N ρ N θ,,y N Θθ,,y ρθ,,y in probabiliy, and o u,,y aifie U2, and he unique diribuion of under hi conrol coincide wih he regular condiional diribuion of given ha Y = y. Thi eablihe 4.9 in he general cae. We reurn now o he pah eimaor wih iniial diribuion µ. The minimizaion of apparen informaion can be expreed in erm of he following conrolled proce wih random iniial condiion: = + b,+a,u, d + σ, dṽ, 4.29 µ.

17 A VARIATIONAL APPROACH TO NONLINEAR ESTIMATION 1829 A imple varian of Lemma 4.1 how ha, if u i coninuou and aifie U2 for all θ R n, hen hi equaion i unique in law and ha a weak oluion for any iniial law, µ. Le P be he diribuion of correponding o he pair µ, u; i follow from 3.14 and he ubequen dicuion ha 4.3 F P,y=F µ, y =h µ µ+ Ju,,y, µ, and hi i minimized by he choice u = u,,y and µ = µ Y,y, where for B B n 4.31 µ Y B,y =P Y χ 1 B,y. Thu, for each y, he regular condiional probabiliy diribuion P Y,y i Markovian wih iniial marginal µ Y,y and differenial operaor 4.32 L y = i b + au,,y i z i i,j a i,j 2 z i z j. Of coure, he nonlinear filer and inerpolaor for he proce can be found from he marginal of hi pah pace meaure. 5. The invere problem. The variaional characerizaion of he invere problem par ii and iv of Propoiion 2.1 can alo be applied o he pah eimaor. Thi involve chooing a likelihood funcion o be compaible wih he given regular condiional probabiliy diribuion, P Y,y. In ecion 4, we minimized apparen informaion over probabiliy meaure correponding o weak oluion of Here, we maximize compaible informaion over negaive log-likelihood funcion, H, ha give rie o poerior diribuion of hi ype. Le Ω, F, P, µ, V, and be a defined in ecion 3. For each probabiliy meaure on R n, µ, wih µ µ, and each coninuou u aifying U2 for all θ, le H be a meaurable funcion uch ha d H = log P + K dp 5.1 d µ T = log dµ u σ, dv + 1 T σ u, 2 d + K, 2 where K R and P i a defined following We hall aume ha µ Y,y µ. If hi i no o, hen, a hown in he proof of Propoiion 2.1, we can alway chooe anoher µ reuling in more compaible informaion, for which i i. The erm K in 5.1 i he informaion in he aociaed unpecified obervaion. Inegral log-likelihood funcion of he form 5.1 can be hough of a being aociaed wih obervaion ha are diribued in ime, in ha informaion from hem gradually become available a increae. The characerizaion of P Y in erm of ochaic conrol can be ued o expre he compaible informaion correponding o H a follow: 5.2 G H,y=K H,P Y,y = K + hµ Y,y µ hµ Y,y µ T + u 1 2 u auz,, yp Y χ 1 dz,y d. R n

18 183 SANJOY K. MITTER AND NIGEL J. NEWTON Log-likelihood funcion of he form 5.1 could come from many differen ype of obervaion. The only conrain placed on u here are ha i be coninuou and ha i aify U2 for all θ. We could furher conrain i o ake he form uz, = Dṽ z,,ỹ, where T ṽz,,ỹ = log E exp ỹ 1 2 gz, g z, d for appropriae g and ỹ. Thi would correpond o obervaion of he ignal-pluwhie-noie variey imilar o 3.2 bu wih conrolled obervaion funcion and pah, g and ỹ. Thi would how he effec of error in he obervaion funcion or approximaion of he obervaion pah. Under appropriae regulariy condiion, ṽ will aify he following parial differenial equaion: ṽ = Lṽ 1 2 DṽaDṽ ỹ g g; ṽ,t=. Thu one inerpreaion of he invere problem involve an infinie-dimenional deerminiic opimal conrol problem in revered ime, wih conrol g, ỹ, and payoff T Π g, ỹ = Dṽa u Dṽ z,, yp Y χ 1 dz,y d. R n The opimal rajecory for hi dual problem, v,,y i a ime-revered likelihood filer for given Y, and he meaure exp vz,, yp χ 1 dz i an unnormalized regular condiional probabiliy diribuion for given obervaion Y Y, T, which coincide wih ha provided by he Zakai equaion for he ime-revered problem. Thi provide an informaion-heoreic explanaion of he connecion beween nonlinear filering and ochaic opimal conrol ued in [6] a well a widening i cope. For a omewha differen problem involving opimizaion over obervaion funcion, ee [16]. 6. Informaion flow and localizaion. The reul of ecion 2 concerning he informaion conerving properie of Bayeian eimaor can be localized in he conex of he diffuion problem 3.1, 3.2. Propoiion 2.1 can be applied o provide variaional characerizaion of variou condiional probabiliie of he pah meaure P Y, including raniion probabiliie, and hee can be ued o characerize he flow of informaion a a given ime and in a given ae. For any iniial law µ µ and any conrol u aifying U1 and U2 for all θ, le Ω, F, F, P,,Ṽ be a weak oluion of 4.29, le P be he diribuion of, and le P,, P,, and P, Y,y be he rericion of P, P, and P Y,y o a defined in 3.4. I follow from he reul of ecion 4 ha P, Y,y coincide wih P, when µ = µ Y,y and u,=u,,y for. A hown in he dicuion following 3.13, hi i he unique probabiliy meaure on minimizing he apparen informaion The um of he fir wo erm on he righ-hand ide of 3.14 i he apparen informaion of P, in he conex of eimaor of, given obervaion Y,, which we can hink of a being he apparen informaion up o ime. The hird erm on he righ-hand ide

19 A VARIATIONAL APPROACH TO NONLINEAR ESTIMATION 1831 of 3.14 i he informaion in he obervaion Y Y, T, which we can hink of a being he informaion remaining in he obervaion Y a ime. A increae, he eimaor correponding o µ, u progreively conver obervaion informaion ino apparen informaion. If u = u,,y, hen hi proce i conervaive, in ha F P,,y doe no change wih. However, if u i no opimal, hen he apparen informaion can increae faer han he obervaion informaion decreae. We can refine hi argumen a follow. Le d Ĩ = log P, 6.1 dp + H p,,,,y+v,,y for T,, where H p i defined in 3.1. Then i follow from 3.11 ha, for all T, d Ĩ = Ĩ + log P, dp, dp, d P + H p,,, r, r T,y, v,,y v,,y. Le Q and Q be, repecively, he diribuion of r z,, r a defined in ecion 3 wih and wihou he applicaion of he conrol ur z,,r, r. The apparen informaion of Q in he conex of eimaor for r z,, r given Y z, i F, z, Q,y=h Q Q + H p,, z,,y, Q + vχ,,y, Q, 6.3 = vz,, y+ 1 σ u u z,r,y 2 Q χ 1 r d z dr, 2 R n where we have ued 2.1. I now follow ha ẼĨ F =Ĩ + 1 Ẽ σ u u 2 r,r,y 2 F dr. Thu Ĩ, F i a ubmaringale and a maringale if u = u,,y. Thi i he Davi Varaiya [4] characerizaion of he opimal conrol for he problem of ecion 4. Seing = + δ in 6.3, we obain he following local informaion quaniie: 6.4 h Q Q = 1 2 σ uz, 2 δ + oδ, 6.5 H p, + δ, z,,y, Q = gz δy gz 2 δ + oδ, 6.6 vχ +δ,+ δ, y, Q = vz,, y+gz δy u 1 2 u au g 2 z,, yδ + oδ. Equaion 6.4 how he local increae in informaion gain of he diribuion of he proce 4.29 over P, 6.5 how he local increae in he reidual informaion of he eimaor P, and 6.6 how he local decreae in he average informaion remaining in he obervaion afer ime. Ify i differeniable a, hen here i a local rae of increae of apparen informaion of σ uz, 2 /2 ẏ g/2 gz and a local rae of

20 1832 SANJOY K. MITTER AND NIGEL J. NEWTON decreae of he remaining obervaion informaion of u u /2 au z,, y ẏ g/2 gz. The former exceed he laer unle he conrol i opimal. The dual problem can alo be localized in hi way. For u a above, le H p be a meaurable funcion uch ha 6.7 H p,, z, z, = u σ z, r,r dv r K K, σ ur z,,r 2 dr where K i differeniable and K T =. Thi can be hough of a being he equivalen of H p,, z, z,,y for an unpecified ime-diribued obervaion uch ha a ime he remaining informaion in he obervaion i K. Thi correpond o H of 5.1 wih K = K. Le Q be he diribuion of z, r, r when i i conrolled by he opimal conrol. Taking expecaion wih repec o Q in 6.7 and aking he limi a, we obain a local rae of decreae of compaible informaion of u u/2 auz,, y. The local rae of increae of he informaion gain of P Y,y i, of coure, σ u z,, y 2 /2. The laer exceed he former unle u i opimal. In he global dual problem 5.1, he regular condiional probabiliy P Y,y i he ource of informaion. A ime he informaion in hi ource i S = h µ µ+ 1 σ u z,, y 2 P Y χ 1 dz,y d. 2 R n A ime T here i no informaion in he obervaion and no reidual informaion all he informaion i ill in he ource. A decreae, informaion flow ou of he ource a a rae Ṡ; i i merged wih reidual informaion and flow ino he obervaion a a rae K.Ifui opimal, hen he flow i conervaive, wherea more generally informaion i lo. Le H z, be he Hilber pace of n-vecor of real wih inner produc α, β z, = α az,β. The developmen above how ha he regular condiional probabiliy P Y,y i locally characerized a he poin z, by he diffuion coefficien az, and bz,+az,α, where α minimize α 2 z, α, u z,, y z,, wherea he opimal rajecory in he dual problem 5.3 i locally characerized in ha i negaive gradien a he poin z,, β maximize 6.9 β,u z,, y z, 1 2 β 2 z,. The local balance of he Bayeian pah eimaor i hu characerized by he Legendre ranform pair 6.8, 6.9. Of coure, hi i he characerizaion of he opimal conrol problem of ecion 4 provided by he ochaic maximum principle, he adjoin proce being he gradien of he opimal dual ae, v,,y, evaluaed a,.

21 A VARIATIONAL APPROACH TO NONLINEAR ESTIMATION Concluion. Thi aricle ha developed dual variaional characerizaion of Bayeian eimaion, in which he co funcional have paricular informaionheoreic meaning. Thee characerizaion provide a naural framework for he udy of modeling and approximaion error in eimaor uch a nonlinear filer. They alo link uch iue wih a broader heory of ochaic diipaivene ee [1], on which he idea and echnique of aiical phyic can be brough o bear. We believe ha hi will have a number of advanage, for example, in he udy of he long-erm behavior of ochaic yem. For a recen developmen of hi ype ee [12]. The characerizaion alo provide a framework for he repreenaion of eimaor, in a broader conex, a apparen informaion minimizer and compaible informaion maximizer. Thee iue will be explored elewhere. Acknowledgmen. The auhor would like o hank one of he referee and he Aociae Edior for heir commen and uggeion which have led o ubanial improvemen in he paper. The econd auhor would like o hank Deni Talay and Francoi LeGland for heir hopialiy during hi vii o he INRIA reearch group Omega and Sigma2. REFERENCES [1] V. S. Borkar and S. K. Mier, A noe on ochaic diipaivene, in Direcion in Mahemaical Syem Theory and Opimizaion, A. Ranzer and C. I. Byrne, ed., Springer- Verlag, NewYork, 22, pp [2] J. M. C. Clark, The deign of robu approximaion o he ochaic differenial equaion of nonlinear filering, in Communicaion Syem and Random Proce Theory, NATO Advanced Sudy In. Ser., E: Appl. Sci. 25, J. K. Skwirzynki, ed., Sijhoff and Noordhoff, Alphen aan den Rijn, 1978, pp [3] M. H. A. Davi, A pahwie oluion of he equaion of nonlinear filering, Theory Probab. Appl., , pp [4] M. H. A. Davi and P. P. Varaiya, Dynamic programming condiion for parially obervable ochaic yem, SIAM J. Conrol, , pp [5] P. Dupui and R. S. Elli, A Weak Convergence Approach o he Theory of Large Deviaion, Wiley, NewYork, [6] W. H. Fleming and S. K. Mier, Opimal conrol and nonlinear filering for nondegenerae diffuion procee, Sochaic, , pp [7] A. Friedman, Sochaic Differenial Equaion and Applicaion, Vol. 1, Academic Pre, New York, [8] H.-O. Georgii, Gibb Meaure and Phae Traniion, de Gruyer, Berlin, [9] R. S. Liper and A. N. Shiryayev, Saiic of Random Procee 1 General Theory, Springer-Verlag, NewYork, [1] T. Mikami, Dynamical yem in he variaional formulaion of he Fokker Planck equaion by he Waerein meric, Appl. Mah. Opim., 42 2, pp [11] S. K. Mier and N. J. Newon, The dualiy beween eimaion and conrol, in Opimal Conrol and Parial Differenial Equaion, J. L. Menaldi, E. Rofman, and A. Sulem, ed., IOS Pre, Amerdam, 2. [12] S. K. Mier and N. J. Newon, Informaion flow and enropy producion in he Kalman Bucy filer, ubmied. [13] N. J. Newon, Obervaion ampling and quaniaion for coninuou-ime eimaor, Sochaic Proce. Appl., 87 2, pp [14] J. Picard, Robuee de la oluion de probleme de filrage avec brui blanc independan, Sochaic, , pp [15] L. C. G. Roger and D. William, Diffuion, Markov Procee and Maringale: Par 2 Iô Calculu, Wiley, NewYork, [16] B. M. Miller and W. J. Runggaldier, Opimizaion of obervaion: A ochaic conrol approach, SIAM J. Conrol Opim., , pp