Information and estimation in Fokker-Planck channels

Transcription

1 Informaion an esimaion in Fokker-Planck channels Anre Wibisono, Varun Jog, an Po-Ling Loh Deparmens of Elecrical & Compuer Engineering an Saisics Universiy of Wisconsin - Maison Maison, WI aywibisono@wisc.eu, vjog@wisc.eu, loh@ece.wisc.eu Absrac We suy he relaionship beween informaion- an esimaion-heoreic quaniies in ime-evolving sysems. We focus on he Fokker-Planck channel efine by a general sochasic ifferenial equaion, an show ha he ime erivaives of enropy, KL ivergence, an muual informaion are characerize by esimaion-heoreic quaniies involving an appropriae generalizaion of he Fisher informaion. Our resuls vasly exen De Bruijn s ieniy an he classical I-MMSE relaion. I. INTODUCTION Informaion heory an saisical esimaion are closely inerwine. Various ieniies an inequaliies arise from funamenal conceps such as muual informaion, Fisher informaion, an esimaion error, an close parallels beween he fiels provie an avenue for evising an eriving new resuls. As a canonical example, he informaion-esimaion resul known as he momen-enropy inequaliy [] saes ha among all coninuous ranom variables wih a fixe variance, Gaussian ranom variables maximize enropy. Furhermore, Sam s inequaliy [2] saes ha for a fixe Fisher informaion, Gaussian ranom variables minimize enropy. The celebrae Cramér-ao boun from saisics [3], which esablishes a lower boun on he variance of an esimaor in erms of he Fisher informaion, follows from he aforemenione facs. In he pas ecae, significan effor has been evoe o uncovering new relaionships beween informaion-heoreic an esimaion-heoreic quaniies, beginning wih he I-MMSE ieniy of Guo e al. [4] for aiive Gaussian noise channels. The ieniy, provie in Theorem below, saes ha he erivaive of he muual informaion beween he channel inpu an oupu wih respec o he signal o noise raio (snr) is proporional o he minimum mean square error (mmse) in esimaing he inpu from he oupu. As erive in Sam [2], he resul is equivalen o De Bruijn s ieniy (cf. equaion (2)). The resaemen of De Bruijn s ieniy in erms of he mmse spawne a hos of aiional informaion-esimaion resuls, incluing an exension o non-gaussian aiive noise [5], a generalizaion o he mismache esimaion seing [6], an several poinwise informaion-esimaion relaions [7]. Differen I-MMSE ype relaions were also obaine for he Poisson channel [8], [9] an Lévy channel [0]. In informaion heory, a channel is a coniional isribuion relaing inpu symbols o oupu symbols. Whereas his coing-heoreic moel is very useful for communicaion channels, however, i possesses cerain rawbacks. Many realworl examples such as weaher sysems an financial markes are bes explaine as sysems evolving in ime accoring o ranom an eerminisic influences. Alhough i is possible o view an evolving sysem as a communicaion channel, where he curren sae is he channel inpu an he sae a a fuure ime is he channel oupu, such an inerpreaion lacks insigh abou he pah of he sysem. Noneheless, informaionheoreic ieas are sill useful in characerizing he behavior of ime-evolving sysems. For insance, one migh characerize how much informaion abou he fuure is conaine in he presen sae using quaniies such as enropy, KL ivergence, an muual informaion. This has been iscusse exensively in he climae science lieraure [], [2], [3], [4]. One of he simples an mos useful ways of moeling evolving sysems is via coninuous-ime Markov chains wih a coninuous sae space, which may be analyze using sochasic ifferenial equaions (SDEs). In paricular, he probabiliy isribuion of such sysems evolves accoring o a parial ifferenial equaion known as he Fokker-Planck equaion. We focus on an informaion-heoreic analysis of ime-evolving sysems escribe by SDEs, an suy he rae of change of various funamenal quaniies as a funcion of ime. We show ha hese raes are convenienly expresse in erms of a generalize Fisher informaion, so our resuls may be inerpree as generalizaions of De Bruijn s ieniy for he SDE or Fokker-Planck seing. Noably, we obain a clean ieniy expressing he ime erivaive of muual informaion in erms of he muual Fisher informaion, allowing us o erive new I- MMSE relaions. Our resuls are reaily specialize o specific sochasic processes, incluing Brownian moion, Ornsein- Uhlenbeck processes, an geomeric Brownian moion. The remainer of he paper is organize as follows: In Secion II, we review exising resuls an presen he family of SDEs o be suie in our paper, as well as he associae Fokker-Planck equaion. In Secion III, we evelop our main resuls concerning he evoluion of enropy, KL ivergence, an muual informaion in erms of esimaion-heoreic quaniies. We conclue in Secion IV wih a iscussion of open quesions. For rigorous proofs of all our resuls, see he exene version of our paper [5].

2 II. BACKGOUND AND POBLEM SETUP We begin by presening formal saemens of De Bruijn s ieniy an he I-MMSE relaion, followe by a eaile characerizaion of he SDE framework iscusse in our paper. A. DeBruijn s ieniy an I-MMSE Consier he aiive Gaussian noise channel X = X 0 + Z () where Z N (0, ) is inepenen of X 0, an > 0 is a ime parameer in his case equal o he variance of he noise ha conrols how much ranomness is ae o he sysem. As increases, we expec he oupu X o be more ranom. De Bruijn s ieniy [2] confirms his inuiion an assers ha H(X ) = 2 J(X ), (2) where H(X )= p (x) log p (x)x is he Shannon enropy wih p enoing he ensiy of X, an J(X ) = E [ ( x log p (X ) ) ] 2 = p (x) 2 x > 0 (3) p (x) is he (nonparameric) Fisher informaion. Anoher common parameerizaion of he channel () is Y snr = snrx + Z where snr > 0 is he signal o noise raio an Z N (0, ) is inepenen of X. Guo e al. [4] esablishe he following I-MMSE relaion, which saes ha he muual informaion I(snr) = I(X; Y snr ) increases a a rae given by he mmse, an showe heir resul is equivalen o De Bruijn s ieniy (2). Theorem (Guo e al. [4]). We have snr I(snr) = 2 mmse(x Y snr), where mmse(x Y snr ) = E[(X E[X Y snr ]) 2 ] enoes he minimum mean square error for esimaing X from Y snr. In erms of he ime parameerizaion (), by seing snr = an X = Y / we see ha Theorem is equivalen o I(X 0; X ) = 2 2 mmse(x 0 X ). (4) B. SDEs an Fokker-Planck equaion Consier a general channel ha oupus a real-value sochasic process (X ) 0 following he SDE X = a(x, ) + σ(x, )W, (5) where (W ) 0 is sanar Brownian moion, an a(x, ) an σ(x, ) > 0 are arbirary real-value smooh funcions. The choice a 0 an σ generaes he Gaussian channel (), bu he SDE framework (5) is consierably more general. The rif a(x, ) is he eerminisic par of he ynamics, an he iffusion σ(x, )W inrouces ranomness by incremenally aing Gaussian noise: For small δ > 0, we have X +δ X + a(x, )δ + σ(x, ) δz where Z N (0, ) is inepenen of X. A convenien way o suy X is via is ensiy p. When X follows he SDE (5), he ensiy funcion p(x, ) = p (x) saisfies a parial ifferenial equaion calle he Fokker-Planck equaion. This resul is classical an echnically requires ha a(x, ) an σ(x, ) saisfy appropriae regulariy an growh coniions (e.g., smoohness an Lipschiz properies [6]). Lemma. The ensiy p(x, ) of he process escribe by equaion (5) saisfies = x (a(x, )p(x, )) + (b(x, )p(x, )), (6) 2 x2 where b(x, ) = σ(x, ) 2. Noe ha Lemma implies he channel (5) is linear in he space of inpu isribuions, which means a mixure of inpus prouces a mixure of oupus. Also noe ha when b(x, ) is inepenen of x, our moel falls uner he PDE framework of Toranzo e al. [7]; however, our SDE framework (5) is somewha more general. Consier he following examples: ) Brownian moion: As iscusse above, he choice a 0 an σ prouces he soluion X = X 0 + W = X0 + Z N (X0, ), which is precisely he Gaussian channel (). The Fokker-Planck equaion is he classical hea equaion = 2 p(x, ) 2 x 2. Any saring poin X 0 = x 0 gives rise o he explici soluion p(x, ) = exp ( (x x 0) 2 ). (7) 2π 2 2) Ornsein-Uhlenbeck process: The SDE is 2 X = αx + W, wih α > 0. This correspons o a(x, ) = αx an σ(x, ) = b(x, ). The Ornsein-Uhlenbeck process is mean-revering an arises in sochasic moeling of ineres raes ( an paricle velociies. The explici soluion is X = e α X 0 + X N (0, ) = 0 eαs W s e α X 0 + ( e )Z, so ) as. The Fokker-Planck equaion is = α x (xp(x, )) + 2 p(x, ) 2 x 2, (8) which may also be solve explicily [8]. Noe ha if α 0, we recover Brownian moion (7). 3) Geomeric Brownian moion: The SDE is X = µx + σx W, where µ an σ > 0. This correspons o a(x, ) = µx, σ(x, ) = σx, an b(x, ) = σ 2 x 2. Geomeric Brownian moion is use o moel asse prices in financial mahemaics, noably in Black-Scholes opion pricing [9]. The explici soluion ( is X N log X 0 + (µ σ2 2 ), σ2 = X 0 exp((µ ) σ2 2 ) + σw ), so log X. The Fokker-Planck equaion is = µ σ2 2 (xp(x, )) + x 2 x 2 (x2 p(x, )), (9) which also has an explici soluion [8].

3 III. MAIN ESULTS We now generalize he informaion-esimaion relaions for he Gaussian channel () o he SDE channel (5). We express he ime erivaives of funamenal informaionheoreic quaniies enropy, relaive enropy, an muual informaion in erms of a generalize Fisher informaion, proucing analogs of De Bruijn s ieniy (2). Alhough enropy is no always increasing, relaive enropy an muual informaion ecrease a a rae given by he relaive an muual Fisher informaion, respecively. We furher inerpre he Fisher informaion via a generalize Bayesian Cramer-ao lower boun an express he muual Fisher informaion as he mmse of esimaing a funcion of he inpu from he oupu, hus proucing generalizaions of he I-MMSE relaion (4). A. Generalize Fisher informaion Le b: (0, ) be a posiive funcion. We efine he Fisher informaion wih respec o b o be [ ( ) ] 2 J b (X)=E b(x) X log p(x) = b(x) p (x) 2 x, (0) where p enoes he isribuion of he ranom variable X. When b, we obain he usual Fisher informaion (3). Our generalize Fisher informaion iffers from oher noions in lieraure in ha we are sill moivae by Shannon enropy. In conras, Toranzo e al. [7] consier φ-fisher informaion o suy φ-enropy, while Luwak e al. [20] an Bercher [2] consier q-fisher informaion o suy q-enropy, which inclues ényi an Tsallis enropies. For isribuions p an q, we efine he relaive Fisher informaion wih respec o b: ( ) 2 J b (p q) = b(x) log x, () x q(x) Alernaively, J b (p q) may be viewe as he Bregman ivergence of J b. ecall ha he KL ivergence, or relaive enropy, may be wrien as he Bregman ivergence of Shannon enropy. Lemma 2. J b (p q) = J b (p) J b (q) J b (q), p q. In paricular, J b (p q) 0 shows ha J b (p) is convex in p. Analogous o he he muual informaion, which measures he reucion in coniional enropy, we efine he muual Fisher informaion of ranom variables X an Y wih respec o b: J b (X; Y ) = J b (Y X) J b (Y ), (2) where J b (Y X) = p X(x)J b (Y X = x) y is he coniional Fisher informaion. Noe ha J b (X; Y ) is he ifference beween E[J b (p Y X ( X))] an J b (E[p Y X ( X)]) = J b (Y ). Since J b is convex, his implies J b (X; Y ) 0. B. From informaion o esimaion In fac, he muual Fisher informaion (2) is equal o a naural generalizaion of he saisical Fisher informaion, a cenral esimaion-heoreic quaniy. ecall ha he poinwise saisical Fisher informaion Φ(y) = Φ(X Y = y) of a parameerize family of isribuions {p X Y ( y)} is 2x. Φ(X Y = y)= p X Y (x y)( y log p X Y (x y)) (3) We efine he saisical Fisher informaion wih respec o b as he weighe average of he poinwise Fisher informaion: Φ b (X Y ) = p Y (y) b(y) Φ(X Y = y) x. (4) The following key resul provies a brige beween informaion an esimaion: Theorem 2. The muual Fisher informaion is equal o he saisical Fisher informaion: J b (X; Y ) = Φ b (X Y ). We now erive wo heorems illusraing he inimae connecions beween he saisical Fisher informaion an quaniies in esimaion heory. ) Esimaion-heoreic lower boun: The poinwise Fisher informaion (3) has a naural esimaion inerpreaion via he Cramer-ao lower boun; similarly, he saisical Fisher informaion (4) provies a lower boun on he esimaion error when we have a prior on he parameers. The following resul is a weighe version of van Trees inequaliy [22]: Theorem 3. Consier a parameerize family of isribuions {p Y X ( x)} wih prior p X (x). For any esimaor T (y) of x, [ ] E (T (Y ) X)2 b(x) Φ b (Y X) + J b (X) = J b (X Y ). Thus, he coniional Fisher informaion J b (X Y ) is inversely proporional o he harness of esimaing X from Y. 2) mmse relaion: We efine he mmse of Y given X wih respec o b as mmse b (Y X) = min T E[b(X)(T (X) Y )2 ], where he minimizaion is over all esimaors T (X). Noe ha he minimizer correspons o he coniional expecaion T (X) = E[Y X], regarless of b. For a paramerize family of isribuions {p Y X ( x)}, consier he poinwise score funcion ϕ(x, y) = y log p Y X(y x). Given a prior p X, by Bayes rule we can efine he oher coniional isribuion p X Y (x y) p X (x)p Y X (y x). Observe ha for every fixe y, if X p X Y ( y), hen ϕ(x, y) is an unbiase esimaor of he nonparameric score funcion: E[ϕ(X, y) Y = y] = y log p Y (y). This fac leas o he following resul: Theorem 4. J b (X; Y ) = mmse b (ϕ(x, Y ) Y ). C. De Bruijn s ieniy We now escribe our generalizaions of De Bruijn s ieniy. In he saemens below, we wrie b (x) = b(x, ) = σ(x, ) 2.

4 ) Time erivaive of enropy: Our firs resul relaes he rae of change of Shannon enropy o he Fisher informaion: Theorem 5. Le X be he oupu of he SDE (5). Then H(X ) = [ 2 J b (X ) + E x a(x, ) 2 ] 2 x 2 b(x, ). Noe ha in he case of hea equaion, when a 0 an b, his resul recovers he classical De Bruijn s ieniy (2). However, in he general case, he enropy oes no necessarily always increase. This may seem o, bu as he resuls below show, we obain monooniciy by consiering relaive enropy. 2) Time erivaive of KL ivergence: Le K(p q) = log q(x) x enoe he KL ivergence. The following resul esablishes ha he relaive enropy beween any wo soluions is always ecreasing, wih a rae given by he relaive Fisher informaion: Theorem 6. Le X, Y enoe he oupu ranom variables of he channel (5) wih isribuions p, q. Then K(p q ) = 2 J b (p q ). Thus, he KL ivergence is a conracing map along any wo rajecories p an q, implying he exisence of a mos one fixe poin of he channel; i.e., a saionary measure p saisfying equaion (6). However, such a isribuion p may no exis, or i may no be a proper probabiliy isribuion (e.g., he Lebesgue measure in he case of he hea equaion). 3) Time erivaive of muual informaion: ecall ha he muual informaion saisfies I(X; Y ) = H(Y ) H(Y X), where H(Y X) = p X(x)H(Y X = x) x is he coniional Shannon enropy. Consier he ime erivaive of I(X 0 ; X ), where X is he oupu of he channel (5) wih inpu X 0. Theorem 5 expresses he ime erivaive of H(X ) in erms of he Fisher informaion J b (X ); since he channel (5) is linear, we also obain a formula for he ime erivaive of he coniional enropy H(X X 0 ) in erms of J b (X X 0 ). ecalling efiniion (2), his yiels he following resul: Theorem 7. The oupu X of he SDE channel (5) wih inpu X 0 saisfies I(X 0; X ) = 2 J b (X 0 ; X ). D. Special cases We specialize our resuls o he examples in Secion II-B. Deails are provie in he exene version of our paper [5]. ) Brownian moion: Since b, he funcion J b is he usual nonparameric Fisher informaion J, an Theorem 5 yiels De Bruijn s ieniy (2). We calculae he coniional Fisher informaion using he coupling X = X0 + Z: J(X X 0 ) = p X0 (x 0 ) J(x 0 + Z) x 0 =. The muual Fisher informaion is Φ(X 0 X ) = J(X ) 0, which implies J(X ). Theorem 7 says ha I(X 0; X ) = ( J(X ) ). 2 Using he formula (7), we may compue he score funcion: ϕ(x 0, x ) = By Theorem 4, we hen obain x log p X X 0 (x x 0 ) = (x x 0 ). Φ(X 0 X ) = 2 mmse((x X 0 ) X ) = 2 mmse(x 0 X ), which, wih Theorem 7, recovers he I-MMSE ieniy (4). 2) Ornsein-Uhlenbeck process: Again, we have J b = J. Applying Theorem 5 yiels H(X ) = 2 J(X ) α. Noe ha if α 0, enropy always increases bu if α > 0, which is he regime of ineres, enropy nee no be monoonic. Using X = e α (X 0 + (e )Z), we may compue he coniional Fisher informaion J(X X 0 ) = e an he muual Fisher informaion Φ(X 0 X ) = e J(X ). Since Φ(X 0 X ) 0, his yiels he boun J(X ) e, which monoonically ecreases o as. Using he explici soluion o equaion (8), we obain ϕ(x 0, x ) = (x e α x 0 ) e. By Theorems 4 an 7, we hen euce he I-MMSE relaion I(X 0; X ) = 2 e ( e ) 2 mmse(x 0 X ). 3) Geomeric Brownian moion: Since b(x, ) = σ 2 x 2, we have J b J. Applying Theorem 5 yiels H(X ) = 2 J b(x ) + µ 2 σ2. Thus, he enropy may no increase monoonically. Using he explici soluion o equaion (9), we may compue ϕ(x 0, x ) = ( ) log x log x 0 α x σ 2 +. Then by Theorem 4, we erive he I-MMSE relaion I(X 0; X ) = 2σ 2 2 mmse(log X 0 X ). E. Mulivariae exension Our resuls exen wihou ifficuly o he mulivariae seing where X is a sochasic process in evolving accoring o he SDE (5), where a(x, ) is a rif vecor, σ(x, ) is a covariance marix, an W is sanar Brownian moion in. The weigh marix is given by b(x, ) = σ(x, )σ(x, ), which is assume o be uniformly posiive efinie.

5 We efine he generalize Fisher informaion (0) wih respec o a posiive efinie marix b(x): [ ] 2 J b (X) = E log p(x) 2 b(x) b(x) = x, where v 2 b(x) = v b(x)v = Tr(b(x)vv ) is he Mahalanobis inner prouc of v. The relaive () an muual Fisher informaion (2) are efine similarly, an he saisical Fisher informaion (4) is efine as Φ b (X Y ) = p Y (y) Tr(b(y)Φ(X Y = y)) y, where Φ(X Y = y) is he usual Fisher informaion marix p X Y (x y)( y log p X Y (x y))( y log p X Y (x y)) x. Wih hese efiniions, all our resuls hol unchange (see he exene version of our paper [5] for more eails). IV. DISCUSSION AND FUTUE WOK We have esablishe informaion-esimaion ieniies for ime-evolving sysems. Our resuls exen he classical De Bruijn s ieniy, which concerns he rae of enropy growh in a Brownian moion process, o he ime erivaives of enropy, KL ivergence, an muual informaion for processes escribe by general SDEs. The preicabiliy of such sysems relies on he informaion conaine in he curren sae regaring fuure saes. A a high level, he curren sae conains progressively less informaion abou fuure saes of he sysem; we erive he specific raes of change in informaion in erms of quaniies arising from esimaion. Theorem 7 relaes he ime erivaive of I(X 0 ; X ) o he saisical Fisher informaion, which by Theorem 3 is inversely proporional o he ifficuly of esimaing X 0 from X. As ime increases, his ifficuly shoul be increasing, suggesing ha I(X 0 ; X ) may ecrease in a convex manner. In he Gaussian seing, Cosa [23] showe ha he enropy of Brownian moion is a convex funcion, an Chen e al. [24] showe ha he firs four erivaives of enropy alernae in sign. We conjecure a similar propery for higher-orer erivaives of muual informaion in general SDEs. We also suspec ha he relaionship beween Fisher informaion an esimaion may be generalize o Bregman ivergences. Our resuls are base on he quaraic funcion f x (y) = 2 b(x)y2, corresponing o Gaussian ranomness generae by he Brownian moion b(x)w. I may be ineresing o invesigae sochasic processeses corresponing o general convex funcions. A final irecion is o explore he connecion o opimal ranspor, which inerpres he Fokker-Planck equaion (6) as he graien flow of relaive enropy in he space of probabiliy ensiies wih respec o he Wassersein meric. Villani [25] iscusses his in eail an inerpres Theorem 6 as he analog of De Bruijn s ieniy, since boh may be viewe as enropy proucion inequaliies. Noe ha such a viewpoin focuses on a PDE raher han SDE formulaion. EFEENCES [] T. M. Cover an J. A. Thomas, Elemens of informaion heory. John Wiley & Sons, 202. [2] A. Sam, Some inequaliies saisfie by he quaniies of informaion of Fisher an Shannon, Informaion an Conrol, vol. 2, no. 2, pp. 0 2, 959. [3] E. L. Lehmann an G. Casella, Theory of poin esimaion. Springer Science & Business Meia, [4] D. Guo, S. Shamai, an S. Verú, Muual informaion an minimum mean-square error in Gaussian channels, IEEE Transacions on Informaion Theory, vol. 5, no. 4, pp , [5] D. P. Palomar an S. Verú, epresenaion of muual informaion via inpu esimaes, IEEE Transacions on Informaion Theory, vol. 53, no. 2, pp , [6] S. Verú, Mismache esimaion an relaive enropy, IEEE Transacions on Informaion Theory, vol. 56, no. 8, pp , 200. [7] K. Venka an T. Weissman, Poinwise relaions beween informaion an esimaion in Gaussian noise, IEEE Transacions on Informaion Theory, vol. 58, no. 0, pp , 202. [8]. Aar an T. Weissman, Muual informaion, relaive enropy, an esimaion in he poisson channel, IEEE Transacions on Informaion heory, vol. 58, no. 3, pp , 202. [9] J. Jiao, K. Venka, an T. Weissman, Poinwise relaions beween informaion an esimaion in he poisson channel, in 203 IEEE Inernaional Symposium on Informaion Theory, pp , IEEE, 203. [0] J. Jiao, K. Venka, an T. Weissman, elaions beween informaion an esimaion in scalar Lévy channels, in 204 IEEE Inernaional Symposium on Informaion Theory, pp , IEEE, 204. [] L.-Y. Leung an G.. Norh, Informaion heory an climae preicion, Journal of Climae, vol. 3, no., pp. 5 4, 990. [2] T. Schneier an S. M. Griffies, A concepual framework for preicabiliy suies, Journal of Climae, vol. 2, no. 0, pp , 999. [3]. Kleeman, Measuring ynamical preicion uiliy using relaive enropy, Journal of he Amospheric Sciences, vol. 59, no. 3, pp , [4] T. DelSole, Preicabiliy an informaion heory. par I: Measures of preicabiliy, Journal of he Amospheric Sciences, vol. 6, no. 20, pp , [5] A. Wibisono, V. Jog, an P.-L. Loh, Informaion an esimaion in Fokker-Planck channels, arxiv preprin, 207. [6] M. C. Mackey, Time s Arrow: The Origins of Thermoynamics Behavior. Springer-Verlag, 992. [7] I. V. Toranzo, S. Zozor, an J.-M. Brossier, Generalizaion of he e Bruijn s ieniy o general φ-enropies an φ-fisher informaions, arxiv preprin arxiv: , 206. [8] S. E. Shreve, Sochasic calculus for finance II: Coninuous-ime moels, vol.. Springer Science & Business Meia, [9] F. Black an M. Scholes, The pricing of opions an corporae liabiliies, The Journal of Poliical Economy, vol. 8, no. 3, pp , 973. [20] E. Luwak, D. Yang, an G. Zhang, epresenaion of muual informaion via inpu esimaes, IEEE Transacions on Informaion Theory, vol. 53, no. 2, pp , [2] J.-F. Bercher, Some properies of generalize fisher informaion in he conex of nonexensive hermosaisics, Physica A: Saisical Mechanics an is Applicaions, vol. 392, no. 5, pp , 203. [22]. D. Gill an B. Y. Levi, Applicaions of he van Trees inequaliy: A Bayesian Cramér-ao boun, Bernoulli, pp , 995. [23] M. Cosa, A new enropy power inequaliy, IEEE Transacions on Informaion Theory, vol. 3, no. 6, pp , 985. [24] F. Cheng an Y. Geng, Higher orer erivaives in Cosa s enropy power inequaliy, IEEE Transacions on Informaion Theory, vol. 6, no., pp , 205. [25] C. Villani, Topics in opimal ransporaion. No. 58, American Mahemaical Soc., 2003.