Bayesian Estimations for Diagonalizable Bilinear SPDEs
BBayesian Estimations for Diagonalizable Bilinear SPDEs
Ziteng Cheng [email protected]
Igor Cialenco [email protected]://math.iit.edu/~igor
Ruoting Gong [email protected]://mypages.iit.edu/~rgong2
Department of Applied Mathematics, Illinois Institute of TechnologyW 32nd Str, John T. Rettaliata Engineering Center, Room 208, Chicago, IL 60616, USA
First Circulated: May 29, 2018This Version: March 1, 2019
Abstract : The main goal of this paper is to study the parameter estimation problem, using theBayesian methodology, for the drift coefficient of some linear (parabolic) SPDEs drivenby a multiplicative noise of special structure. We take the spectral approach by assumingthat one path of the first N Fourier modes of the solution is continuously observed overa finite time interval. First, we show that the model is regular and fits into classicallocal asymptotic normality framework, and thus the MLE and the Bayesian estimatorsare weakly consistent, asymptotically normal, efficient, and asymptotically equivalentin the class of loss functions with polynomial growth. Secondly, and mainly, we prove aBernstein-Von Mises type result, that strengthens the existing results in the literature,and that also allows to investigate the Bayesian type estimators with respect to a largerclass of priors and loss functions than that covered by classical asymptotic theory. Inparticular, we prove strong consistency and asymptotic normality of Bayesian estimatorsin the class of loss functions of at most exponential growth. Finally, we present somenumerical examples that illustrate the obtained theoretical results.
Keywords: statistical inference for SPDEs, Bayesian statistics, Bernstein-Von Mises, parabolicSPDE, multiplicative noise, stochastic evolution equations, identification problems forSPDEs
MSC2010:
The analytical theory for Stochastic Partial Differential Equations (SPDEs) has been extensivelystudied over the past few decades. It is well recognized that SPDEs can be used as an importantmodeling tool in various applied disciplines such as fluid mechanics, oceanography, temperatureanomalies, finance, economics, biological and ecological systems; cf. [Cho07, LR17, LR18]. Onthe other hand, the literature on statistical inference for SPDEs is, relatively speaking, limited.We refer to the recent survey [Cia18] for an overview of the literature and existing methodologieson statistical inference for parabolic SPDEs. Most of the existing results are obtained within theso-called spectral approach, when it is assumed that one path of N Fourier modes of the solution isobserved continuously over a finite interval of time, in which case usually the statistical problemsare addressed via maximum likelihood estimators (MLEs). Asymptotic properties of the estimatorsare studied in the large number of Fourier modes regime, N → ∞ , while time horizon is fixed. Inparticular, there are only few works related to Bayesian statistics for infinite dimensional evolutionequations [Bis02, Bis99, PR00]. As usual, studying SPDEs driven by multiplicative noise is more1 a r X i v : . [ m a t h . S T ] M a r Cheng, Cialenco, and Gong involved, and the parameter estimation problems for such equations are not an exception; theliterature on this topic is also limited [CL09, Cia10, PT07, CH17, BT17].The main goal of this paper is twofold: to study the parameter estimation problem for thedrift coefficient of linear SPDEs driven by a multiplicative noise (of special structure) and byapplying the
Bayesian estimation procedure. Similar to the existing literature, we are assuming thespectral approach, and the main objective is to derive Bayesian type estimators and to study theirasymptotic properties as N → ∞ . Besides contributing to these two important and undevelopedtopics, the obtained results will prepare the foundation for studying similar problems for morecomplex (nonlinear) equations. We consider a multiplicative noise of special structure, which iscustomary considered in the SPDE applied literature. For example, such type of multiplicative noiseappears in modeling and studying the dynamics of geophysical fluids [GHZ09, GHZ08], studyingthe stochastic primitive equations [GHKVZ14] or stochastically forced shell model of turbulent flow[FGHV16], etc. Needless to say, this particular noise structure can be potentially used as modelingfeature in SPDEs where the noise is not derived from the first principles, but rather added tocapture the imperfections in model and/or measurements. This work is the first attempt to studyparameter estimation problems for SPDEs driven by this multiplicative noise. The case of largetime asymptotics is omitted here, since it is easily reduced to the corresponding statistical problemfor finite dimensional stochastic differential equations, which is a well developed field. It is worthmentioning that the asymptotic properties of MLEs for these equations are trivially obtained, andwe mention them here only briefly since they are used in derivation of convergence of proposedBayesian estimators.The main contributions of this paper can be summarized as follows: (cid:5) We derive and study the asymptotic properties of MLE and Bayesian type estimators for thedrift coefficient of a stochastic evolution system driven by a multiplicative (space-time) noise. (cid:5)
We show that the considered statistical model is regular, and uniformly asymptotically nor-mal, in the sense of [IK81], and fits the classical local asymptotic normality (LAN) paradigm.In particular, under suitable assumptions, the MLE and the Bayesian estimators are weaklyconsistent, asymptotically normal, efficient, and asymptotically equivalent in the class of lossfunctions with polynomial growth . (cid:5) We prove a Bernstein-Von Mises type result, that strengthens the existing results in theliterature, and that also allows to investigate the Bayesian type estimators with respect to alarger class of priors and loss functions than that covered by classical asymptotic theory. (cid:5)
We prove strong consistency and asymptotic normality of Bayesian estimators in the class of loss functions of at most exponential growth.
The obtained results and developed techniques, besides their stand along merits, could be poten-tially useful for investigating some related problems, such as asymptotic properties of estimatorsin the simultaneous large times and large number of Fourier modes regime, discrete sampling, etc.The paper is organized as follows. In Section 2 we setup the problem and provide sufficientconditions on model parameters for the well-posedness of the solution of the underlying SPDEs.Also here, we specify the statistical model, and show that the model is regular (in statistical sense)and uniformly asymptotically normal. Section 3 is devoted to MLE and its asymptotic proper-ties. Using LAN approach we show that MLE is weakly consistent, asymptotically normal, andasymptotically efficient; see Theorem 3.2. In addition, we also establish the strong consistency andasymptotic normality of MLE by exploiting the specific structure of the estimators. The Bayesianestimators are investigated in Section 4. We start with the derivation of the Bayesian estimatorsand briefly cite their properties within the existing general inference theory; see Theorem 4.3. Next, ayesian Estimations for SPDEs
Let (Ω , F , { F t } t ≥ , P ) be a stochastic basis satisfying the usual assumptions, on which we considera sequence of independent standard Brownian motions { w k } k ∈ N . Assume that H is a separableHilbert space, with the corresponding inner product ( · , · ) H . Let A be a positive definite self-adjointoperator in H that has only point spectrum, denoted by { µ k } k ∈ N , with the corresponding eigen-vectors { h k } k ∈ N . We make the standing assumption that { h k } k ∈ N forms a complete orthonormalsystem in H , and µ k → ∞ . We will denote by { H γ , γ ∈ R } the scale of Hilbert spaces generated bythe operator A , i.e., H γ is equal to the closure of the collection of all finite linear combinations of { h k } k ∈ N with respect to the norm (cid:107) · (cid:107) γ := (cid:107) A γ · (cid:107) H , with A γ v := (cid:80) k ∈ N µ γk v k h k for v = (cid:80) k ∈ N v k h k .We will also denote by [ · , · ] the dual pair between H / and H − / relative to the inner product( · , · ) H .We consider the following stochastic evolution equation (cid:40) d u ( t ) + θAu ( t ) d t = σ (cid:80) ∞ k =1 u k ( t ) h k q k d w k ( t ) ,u (0) = u ∈ H, (2.1)where u k ( t ) := ( u ( t ) , h k ) H , t ≥
0, are the Fourier modes of the solution u with respect to { h k } k ∈ N , θ, σ ∈ R + := (0 , ∞ ), and { q k } k ∈ N is a sequence in R + .The well-posedness of equation (2.1) can be established either directly or by using some standardresults from the general theory of linear SPDEs (see for instance [LR17, Section 4.4]). Theorem 2.1.
Let
T > , u ∈ L (Ω; H ) , and assume that there exist N ∈ N and c > such that θ − σ q k µ k ≥ c, for all k ≥ N . (2.2) Then, equation (2.1) admits a unique solution u ∈ L (Ω; ( C (0 , T ); H )) ∩ L (Ω × (0 , T ); H ) , and E (cid:16) sup t ∈ (0 ,T ) (cid:107) u ( t ) (cid:107) H + (cid:90) T (cid:107) u ( t ) (cid:107) / d t (cid:17) ≤ C (cid:107) u (cid:107) H , for some constant C .Proof. In view of [LR17, Theorem 4.4.3], it is enough to show that the following parabolicitycondition holds true, − θ [ Av, v ] + ∞ (cid:88) k =1 σ q k v k + c A (cid:107) v (cid:107) ≤ M (cid:107) v (cid:107) H , v ∈ H , (2.3)for some positive constant c A and M . Note that, [ Av, v ] = (cid:107) v (cid:107) = (cid:80) ∞ k =1 µ k v k , and thus, the lefthand side of (2.3) writes ∞ (cid:88) k =1 (cid:0) − (2 θ + c A ) µ k + σ q k (cid:1) v k = − ∞ (cid:88) k =1 µ k (cid:18) θ − σ q k µ k − c A (cid:19) v k . Cheng, Cialenco, and Gong
We put c A = c , and by taking into account (2.2), as well as the fact that µ k → ∞ , the condition (2.3)follows at once. Remark . It is worth mentioning two simple examples of system’s parameters that satisfy (2.2):(E1) there exist ε >
0, and
C >
0, such that q k ≤ Cµ − εk , for sufficiently large k .(E2) 2 θ − σ >
0, and q k ≤ µ k , for sufficiently large k .Although the equation (2.1) is driven by a multiplicative noise, due to the special structure ofthe noise, it is a diagonalizable SPDE, namely the Fourier modes of the solution satisfy an infinitedimensional system of decoupled equationsd u k ( t ) + θµ k u k ( t ) d t = σq k u k ( t ) d w k ( t ) , t ∈ [0 , T ] , k ∈ N , (2.4)with initial condition u k (0) = ( u , h k ) H . Hence, we have that u k ( t ) = u k (0) exp (cid:18) − (cid:18) θµ k + 12 σ q k (cid:19) t + σq k w k ( t ) (cid:19) , t ∈ [0 , T ] , k ∈ N . (2.5)Without loss of generality we will assume that u k (0) (cid:54) = 0, for all k ∈ N . We note that, one coulduse directly the above form of u k ’s to prove the well-posedness of (2.1). We will take the continuous-time observation framework by assuming that the solution u , as anobject in H , or in a finite dimensional projection of H , is observed continuously in time for all t ∈ [0 , T ], and for some fixed horizon T . We assume that σ , and q k , k ∈ N , are known constants,since generally speaking under the continuous-time sampling scheme, using quadratic variationarguments, these parameters can be found exactly. We will be interested in estimating the unknownparameter θ ∈ R + , with θ being the true value of this parameter of interest. In what follows wewill denote by u θ the solution to (2.1) that corresponds to the parameter θ , and correspondingly,we put u θk := ( u θ , h k ) H , k ∈ N . If no confusion arises, we will continue to write u and u k instead of u θ and u θ k .We will also assume that q k (cid:54) = 0 for all k ∈ N . If q k = 0 for some k ∈ N , then θ can be foundexactly, and the considered statistical problem becomes trivial.In this study, we will assume that one path of the first N Fourier modes ( u ( t ) , . . . , u N ( t ))is observed continuously over a fixed time interval [0 , T ], for some T >
0. We will focus on theasymptotic properties in large number of Fourier modes, N → ∞ , while T is fixed. The large timeasymptotics T → ∞ , with N fixed, reduces to existing results for finite dimensional systems ofstochastic differential equations, which is well understood. The mixed case with both N, T → ∞ is left for further studies.We begin by placing the considered statistical model within classical asymptotic theory ofstatistical estimation. Let C ([0 , T ]; R N ) denote the space of all R N -valued continuous functionson [0 , T ]. The cylindrical (Borel) σ -field on C ([0 , T ]; R N ) is denoted by B ( C ([0 , T ]; R N )). Forevery θ ∈ R + , let P θN be the probability measure on ( C ([0 , T ]; R N ) , B ( C ([0 , T ]; R N )) induced by theprojected solution U θN := { ( u θ ( t ) , . . . , u θN ( t )) , t ∈ [0 , T ] } . As before, we simply write U N and P N instead of U θ N and P θ N . The measures P θN and P N are equivalent, and the Likelihood Ratio, or theRadon-Nikodym derivative, is given by (cf. [LS00, Section 7.6.4])d P θN d P N ( U N ) = exp (cid:18) θ − θσ N (cid:88) k =1 µ k q − k (cid:90) T d u k ( t ) u k ( t ) + (cid:0) θ − θ (cid:1) T σ N (cid:88) k =1 µ k q − k (cid:19) . (2.6) ayesian Estimations for SPDEs θ is then given by I N = I N ( θ ) := E (cid:32)(cid:18) ∂∂θ ln d P θN d P N ( U N ) (cid:12)(cid:12)(cid:12)(cid:12) θ = θ (cid:19) (cid:33) = Tσ N (cid:88) k =1 µ k q − k , (2.7)which is, in particular, independent of θ .Next we will show that the statistical model E := { C ([0 , T ]; R N ) , B ( C ([0 , T ]; R N )) , P θN , θ ∈ R + } is regular; cf. [IK81, Section I.7]. Theorem 2.3.
The statistical model E is regular in R + .Proof. We will follow [IK81, Section I.7], and show that(a) d P θN / d P N ( U N ) is a continuous function of θ ∈ R + , P -almost surely;(b) { P θN , θ ∈ R + } has finite Fisher information for each θ ∈ R + ;(c) The function ψ ( U N , θ ) := ∂ (cid:113) d P θN / d P N ( U N ) /∂θ is continuous in L (Ω; P ).In view of (2.6) and (2.7), { P θN , θ ∈ R + } clearly satisfies properties (a) and (b). To prove (c), by(2.6) we first note that (cid:113) d P θN / d P N ( U N ) is continuously differentiable with respect to θ in R + , forevery ω ∈ Ω, and ψ ( U N , θ ) = (cid:18) − σ N (cid:88) k =1 µ k q − k (cid:90) T d u k ( t ) u k ( t ) − σ N (cid:88) k =1 µ k q − k (cid:19)(cid:115) d P θN d P N ( U N )is continuous with respect to θ in R + for every ω ∈ Ω. Moreover, for any θ, θ (cid:48) ∈ R + with | θ − θ (cid:48) | ≤ (cid:0) ψ ( U N , θ ) − ψ ( U N , θ (cid:48) ) (cid:1) ≤ (cid:0) ψ ( U N , θ ) + ψ ( U N , θ (cid:48) ) (cid:1) ≤ (cid:18) − σ N (cid:88) k =1 µ k q k (cid:90) T d u k ( t ) u k ( t ) − σ N (cid:88) k =1 µ k q k (cid:19) exp (cid:18) θ + 1 σ N (cid:88) k =1 µ k q k (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) T d u k ( t ) u k ( t ) (cid:12)(cid:12)(cid:12)(cid:12) − θ T σ N (cid:88) k =1 µ k q k (cid:19) , which is integrable with respect to P in view of (2.4). Therefore, by the dominated convergencetheorem, lim θ (cid:48) → θ E (cid:16)(cid:0) ψ ( U N , θ ) − ψ ( U N , θ (cid:48) ) (cid:1) (cid:17) = 0 , which completes the proof of property (c), and thus concludes the proof of the theorem.It turns out that the statistical model E , being regular, fits also nicely in the general frameworkof Local Asymptotic Normality (LAN). As next result shows, E is actually uniformly asymptoticallynormal ; cf. [IK81, Definition II.2.2]). This fundamental property will allow to show that maximumlikelihood estimator and Bayesian estimators for θ are not only consistent and asymptoticallynormal, but also asymptotically efficient and asymptotically equivalent. See Section 3, Section 4and discussions therein on these theoretical aspects, as well as the comparison between them andthose developed and proposed in this paper without using LAN framework. Theorem 2.4.
The family { P θN , θ ∈ R + } is uniformly asymptotically normal. Cheng, Cialenco, and Gong
Proof.
For reader’s convenience, we first match the notations in [IK81, Definition II.2.2] with thosefrom our model: the perturbation variable ε N will be 1 /N , and ϕ ( ε N , t ) = 1 / √ I N . For anysequences { ϑ N } N ∈ N ⊂ R + and { η N } N ∈ N ⊂ R such that η N → η ∈ R , as N → ∞ , and that θ N := ϑ N + η N / √ I N ∈ R + for all N ∈ N , by (2.6) and (2.7) we haved P θ N N d P ϑ N N (cid:0) U ϑ N N (cid:1) = exp (cid:32) ϑ N − θ N σ N (cid:88) k =1 µ k q − k (cid:90) T d u ϑk ( t ) u ϑk ( t ) + (cid:0) ϑ N − θ N (cid:1) T σ N (cid:88) k =1 µ k q − k (cid:33) = exp (cid:18) − η N σ √ I N N (cid:88) k =1 µ k q − k (cid:90) T d u ϑ N k ( t ) u ϑ N k ( t ) − η N − ϑ N η N (cid:112) I N (cid:19) = exp (cid:32) − η N (cid:18) σ √ I N N (cid:88) k =1 µ k q − k (cid:90) T d u ϑ N k ( t ) u ϑ N k ( t ) + ϑ N (cid:112) I N (cid:19) − η (cid:18) η − η N (cid:19)(cid:33) . Clearly η / − η N / →
0, as N → ∞ . It remains to show that η N ξ N converges in distribution tosome centered normal distribution, as N → ∞ , where ξ N := − σ √ I N N (cid:88) k =1 µ k q − k (cid:90) T d u θk ( t ) u θk ( t ) + θ (cid:112) I N = − σ √ I N N (cid:88) k =1 u k q − k w k ( T ) . (2.8)Clearly, ξ N D = N (0 ,
1) under P , and therefore, by Slutsky’s theorem, η N ξ N D −→ N (0 , η ) under P ,as N → ∞ , which completes the proof. In this section, we will investigate the asymptotic properties of the maximum likelihood estimator for the unknown parameter θ . The obtained results, albeit simple, are important on their own, andwe will also use them later to study the Bayesian estimators in Section 4.By maximizing the likelihood ratio in (2.6) with respect to θ , we obtain the following estimatorfor θ (cid:98) θ N := − (cid:80) Nk =1 µ k q − k (cid:82) T u k ( t ) u k ( t ) T (cid:80) Nk =1 µ k q − k . (3.1)Using (2.4), by Itˆo’s formula, the estimator (cid:98) θ N can also be written as (cid:98) θ N := − (cid:80) Nk =1 µ k (cid:0) q − k log( u k ( T ) /u k (0)) + σ T (cid:1) T (cid:80) Nk =1 µ k q − k , (3.2)which can be useful for practical purposes.Since θ ∈ R + , formally, the MLE for θ is given by (cid:98) θ MLE N := { (cid:98) θ N ∈ R + } (cid:98) θ N . (3.3)Before we start the analysis on the asymptotic properties of the MLE, let us first introduce thefollowing classes of loss functions, which will be used in the statement of asymptotic efficiency ofMLE, as well as in the later discussions on Bayesian estimators. Definition 3.1.
Let W be the set of Borel measurable loss functions (cid:96) : R → [0 , ∞ ) such that ayesian Estimations for SPDEs (cid:96) is symmetric on R and is non-decreasing on [0 , ∞ );(ii) (cid:96) (0) = 0 and (cid:96) is continuous at x = 0, but is not identically 0.Denote by W e, the set of functions (cid:96) ∈ W whose growth as | x | → ∞ is bounded by one of thefunctions e c | x | r with c ∈ R + and r ∈ (0 , (cid:96) ∈ W which possess a polynomial majorant as | x | → ∞ will be denoted by W p .We will denote by W (cid:48) the class of loss functions (cid:96) that are non-negative, Borel-measurable, andlocally bounded function on R , with (cid:96) (0) = 0, such that (cid:96) ( x ) ≤ (cid:96) ( x ) , for any x , x ∈ R with | x | ≤ | x | . (3.4)Clearly, W ⊂ W (cid:48) .The following result is a consequence of [IK81, Theorem III.1.1, Theorem III.1.2, CorollaryIII.1.1], which summaries the uniform weak consistency, the uniform asymptotic normality, andthe asymptotic efficiency of (cid:98) θ MLE N . Theorem 3.2.
Let B ⊂ R + be any compact set. Then, the following asymptotic properties of (cid:98) θ MLE N hold true.(a) (cid:98) θ MLE N is weakly consistent, uniformly in θ ∈ B . That is, for any ε > , lim N →∞ sup θ ∈ B P (cid:16)(cid:12)(cid:12)(cid:98) θ MLE N − θ (cid:12)(cid:12) > ε (cid:17) = 0 , (b) (cid:98) θ MLE N is asymptotically normal with parameter ( θ , I − N ) , uniformly in θ ∈ B , where I N isgiven by (2.7) . That is, uniformly in θ ∈ B , (cid:112) I N (cid:0)(cid:98) θ MLE N − θ (cid:1) D −→ N (0 , , N → ∞ . (c) For any ε > , lim N →∞ sup θ ∈ B P (cid:16)(cid:12)(cid:12)(cid:12)(cid:112) I N (cid:0)(cid:98) θ MLE N − θ (cid:1) − ξ N (cid:12)(cid:12)(cid:12) > ε (cid:17) = 0 , where ξ N is defined as in (2.8) .(d) For any (cid:36) ∈ W p , (cid:98) θ MLE N is asymptoticly efficient in B for the loss function (cid:36) N ( x ) := (cid:36) ( √ I N x ) . That is, for any θ ∈ B , lim δ → lim inf N →∞ sup | θ − θ | <δ E (cid:16) (cid:36) N (cid:0)(cid:98) θ MLE N − θ (cid:1)(cid:17) = E ( (cid:36) ( ξ )) , where ξ D = N (0 , under P .Proof. We only need to check the conditions (N1) − (N4) in [IK81, Section III.1]. The condition(N1) follows from Theorem 2.4, while the condition (N2) is trivial since ϕ ( ε, t ) = 1 / √ I N in ourcase. To verify the condition (N3), for any θ ∈ B and η, ζ ∈ R such that η N := θ + η/ √ I N ∈ R + and ζ N := θ + ζ/ √ I N ∈ R + , by (2.6) and (2.8) we have E (cid:115) d P η N N d P N ( U N ) − (cid:115) d P ζ N N d P N ( U N ) = E (cid:18)(cid:16) e − ηξ N ( θ ) / − η / − e − ζξ N ( θ ) / − ζ / (cid:17) (cid:19) . Cheng, Cialenco, and Gong
Similar to the proof of Theorem 2.4, we note that ξ N ( θ ) D = N (0 ,
1) under P , and thus( η − ζ ) − E (cid:115) d P η N N d P N ( U N ) − (cid:115) d P ζ N N d P N ( U N ) = 2 − e − ( η − ζ ) / ( η − ζ ) ≤ . Therefore, the condition (N3) is valid with β = m = 2 and any positive constants B , R , and α .Similarly, for any θ ∈ B , n, N ∈ N , and η ∈ R with η N := θ + η/ √ I N ∈ R + , | η | n E (cid:115) d P η N N d P N ( U N ) = | η | n E (cid:16) e − ηξ N ( θ ) / − η / (cid:17) ≤ sup η ∈ R + | η | n e − η / < ∞ . which verifies the validity of the condition (N4). The proof is complete.Given the particular form of the estimator (cid:98) θ N , one can establish its strong consistency andasymptotic normality. Indeed, by (2.4), (3.1) can be conveniently written as (cid:98) θ N = θ − σT (cid:80) Nk =1 µ k q − k w k ( T ) (cid:80) Nk =1 µ k q − k = θ − σT (cid:80) Nk =1 a k b N , (3.5)where a k := µ n q − n w n ( T ) and b n := (cid:80) nk =1 µ k q − k , n ∈ N . Clearly, b N (cid:37) ∞ as N → ∞ . Moreover,in view of Lemma A.2, we have that ∞ (cid:88) k =1 Var( a k ) b k < ∞ . By the Law of Large Numbers, Theorem A.1, (cid:80) Nk =1 a k /b N → N → ∞ , and thus by (3.5), weobtain that lim N →∞ (cid:98) θ N = θ , P − a.s. , (3.6)namely (cid:98) θ N is a strongly consistent estimator of θ . Together with (3.3), we also have thatlim N →∞ (cid:98) θ MLE N = θ , P − a.s. , that is (cid:98) θ MLE N is a strongly consist estimator of θ . In addition, by (2.7), (2.8), and (3.5), under P , (cid:112) I N (cid:0)(cid:98) θ N − θ (cid:1) = ξ N D = N (0 , , for any N ∈ N and θ ∈ R + . (3.7)Finally, we conclude this section by showing that the estimator (cid:98) θ N is also asymptotically efficientin the class of loss functions with polynomial growth. Proposition 3.3.
The properties (a) − (d) in Theorem 3.2 hold true with (cid:98) θ MLE N replaced by (cid:98) θ N .Proof. The properties (a) − (c) for (cid:98) θ N follow trivially from (3.7). The property (d) for (cid:98) θ N followsfrom (3.7) and [IK81, Theorem III.1.3]. ayesian Estimations for SPDEs In this section, we propose two Bayesian type estimators for θ with respect to scaled and unscaledloss functions, respectively, and we will investigate their asymptotic behavior. First, by similarityto the MLE estimator from previous section and following [IK81, Section III.2], we will study theasymptotic properties of the Bayesian estimator with respect to a scaled loss function (cid:96) N ( · ) := (cid:96) ( √ I N · ), where (cid:96) has a polynomial majorant. However, most of this section will be devotedto results beyond and independent of the classical asymptotic theory. As custom for Bayesianstatistics (cf. [BKPR71] and [Bis02]), we prove a Bernstein-Von Mises theorem for the posteriordensity in which both the prior density and the test function have at most exponential growth rates.Consequently, this allows to study the asymptotic properties of Bayesian estimators with respectto any loss function (cid:96) that has an exponential majorant (in contrast to classical theory that allowsonly polynomial growth). Moreover, we show that the MLE and the proposed Bayesian estimatorsare asymptotically equivalent.We begin with the definition of considered prior densities on R + . Definition 4.1.
Let Q be the set of all non-negative and non-trivial functions on R + . Denote by Q e, the set of functions (cid:37) ∈ Q which are continuous and positive on R + and whose growth as θ → ∞ is bounded by one of the functions e c θ r with c ∈ R + and r ∈ (0 , (cid:37) ∈ Q which are continuous and positive on R + and possess a polynomial majorant as θ → ∞ willbe denoted by Q p .Let (cid:37) ∈ Q . We define the posterior density as p ( θ | U N ) := d P θN d P N ( U N ) (cid:37) ( θ ) (cid:90) R + d P ηN d P N ( U N ) (cid:37) ( η ) d η , θ ∈ R + . (4.1)Informally, as standard in Bayesian inference, we took ‘posterior ∝ likelihood × prior.’ For a moreformal discussion of the rational behind (4.1) we refer to Appendix B.Together with (2.6), (2.7), and (3.1), we deduce that p ( θ | U N ) = exp (cid:18) θ − θσ N (cid:88) k =1 µ k q − k (cid:90) T d u k ( t ) u k ( t ) + (cid:0) θ − θ (cid:1) T σ N (cid:88) k =1 µ k q − k (cid:19) (cid:37) ( θ ) (cid:90) R + exp (cid:18) θ − ησ N (cid:88) k =1 µ k q − k (cid:90) T d u k ( t ) u k ( t ) + (cid:0) θ − η (cid:1) T σ N (cid:88) k =1 µ k q − k (cid:19) (cid:37) ( η ) d η = exp (cid:18) − θσ N (cid:88) k =1 µ k q − k (cid:90) T d u k ( t ) u k ( t ) − θ T σ N (cid:88) k =1 µ k q − k (cid:19) (cid:37) ( θ ) (cid:90) R + exp (cid:18) − ησ N (cid:88) k =1 µ k q − k (cid:90) T d u k ( t ) u k ( t ) − η T σ N (cid:88) k =1 µ k q − k (cid:19) (cid:37) ( η ) d η = e − I N ( θ − (cid:98) θ N ) / (cid:37) ( θ ) (cid:90) R + e − I N ( η − (cid:98) θ N ) / (cid:37) ( η ) d η . (4.2) As usual in statistics, the symbol ∝ will be used to denote equality between two quantities up to a normalizedconstant. Cheng, Cialenco, and Gong
Remark . Clearly both Q e, and Q p are large classes of functions. One class of (conjugate) priorsworth mentioning is the truncated normal N ( µ , σ ; R + ) supported on R + . Let the prior (cid:37) be thedensity of the truncated normal N ( µ , σ ; R + ), namely (cid:37) ( θ ) ∝ φ (( θ − µ ) /σ ) , θ >
0, where φ isthe density of a standard Gaussian random variable. By (4.2), for any θ ∈ R + , we have that p ( θ | U N ) ∝ exp (cid:18) − (cid:0) θ − (cid:98) θ N (cid:1) I − N − ( θ − µ ) σ (cid:19) ∝ exp − (cid:16) θ − (cid:0) σ (cid:98) θ N + I − N µ (cid:1)(cid:14)(cid:0) σ + I − N (cid:1)(cid:17) σ I − N (cid:14)(cid:0) σ + I − N (cid:1) , namely p ( θ | U N ) is a truncated normal N (( σ (cid:98) θ N + I − N µ ) / ( σ + I − N ) , ( σ I − N ) / ( σ + I − N ); R + ).Of course, one can take uninformative prior (cid:37) ( θ ) ∝
1. Although such prior has no probabilisticmeaning, the corresponding posterior is well-defined and preserves all convergence results listedbelow.We now introduce two Bayesian type estimators, one with respect to a loss function (cid:96) ∈ W (cid:48) and one with respect to its scaled version. A Bayesian estimator with respect to (cid:96) ∈ W (cid:48) is definedas (cid:98) β N := arg min β ∈ R + (cid:90) R + (cid:96) ( η − β ) p ( η | U N ) d η, (4.3)given that the minimum is strict and attainable. In line with [IK81, Section III.2], we define aBayesian estimator with respect to (cid:96) N ( x ) := (cid:96) ( √ I N x ), where (cid:96) ∈ W , as (cid:101) β N := arg min β ∈ R + (cid:90) R + (cid:96) (cid:0)(cid:112) I N ( η − β ) (cid:1) p ( η | U N ) d η, (4.4)given that the minimum is strict and attainable.Before proceeding with asymptotic properties of these estimators, several comments pinpointingthe intuition behind these definitions are in order.(a) Let U Θ N be the R N -valued process obtained by substituting θ with Θ in U θN , where Θ :(Ω , F , P ) → ( R + , B ( R + ) admits a proper prior density (cid:37) and is independent of the Brownianmotions { w k , k ∈ N } . Recall that the Bayesian risk of an estimator θ N ∈ σ ( U Θ N ) with respectto a loss function (cid:96) is defined as r (cid:0) θ N ; (cid:96) (cid:1) := E (cid:0) (cid:96) (cid:0) θ N − Θ (cid:1)(cid:1) = E (cid:16) E (cid:16) (cid:96) (cid:0) θ N − Θ (cid:1)(cid:12)(cid:12)(cid:12) U Θ N (cid:17)(cid:17) = E (cid:18) (cid:90) R (cid:96) (cid:0) θ N − η (cid:1) p ( η | U Θ N ) d η (cid:19) , where the last equality follows from (B.2). Hence, (cid:98) β N is the minimizer of the Bayesian risk r ( · ; (cid:96) ), while (cid:101) β N is the minimizer of the Bayesian risk r ( · ; (cid:96) N ).(b) The definition of the Bayesian estimator (cid:98) β N is standard, in which the loss function does notdepend on the sample size N , and is therefore more accessible for computational purposes.On the other hand, as argued in [IK81], using a scaled loss function in defining a Bayesianestimator is more appropriate for analyzing some asymptotic properties of the estimator.Indeed, recall that (cf. [IK81, Definition I.9.1]) a sequence of estimators { θ N } N ∈ N of θ iscalled asymptotically efficient in R + with respect to a sequence of loss function { (cid:36) N } N ∈ N if, for any open set O ⊂ R + and any estimator (cid:101) θ N of θ ,lim N →∞ (cid:18) inf (cid:101) θ N sup θ ∈ O E (cid:16) (cid:36) N (cid:0)(cid:101) θ N − θ (cid:1)(cid:17) − sup θ ∈ O E (cid:16) (cid:36) N (cid:0) θ N − θ (cid:1)(cid:17)(cid:19) = 0 . The asymptotic efficiency studied in part (d) of Theorems 3.2 and 4.3 is a special case under the framework oflocal asymptotic normality. ayesian Estimations for SPDEs (cid:36) N should depend on N (and typically (cid:36) N ( x ) = (cid:36) ( √ I N x )) in order to capture more subtle difference between estimators. Hence,it is natural to scale the loss function in the definition of (cid:101) β N in the same way as (cid:36) N wheninvestigating the asymptotic efficiency of (cid:101) β N .(c) When (cid:96) ( x ) = | x | α for some α > (cid:101) β N coincides with (cid:98) β N . Moreover, when (cid:96) ∈ W p , (cid:101) β N and (cid:98) β N are expected to have the same asymptotic behavior as N → ∞ . Clearly, this may notbe the case if (cid:96) ∈ W e, , or more generally (cid:96) ∈ W (cid:48) , yet another reason to distinguish the twoproposed estimators.The following theorem summarizes the asymptotic properties of (cid:101) β N with (cid:37) , and (cid:96) having poly-nomial growth, which is a direct consequence of [IK81, Theorem III.2.1, Theorem III.2.2]. Theorem 4.3.
Let B ⊂ R + be any compact set, (cid:37) ∈ Q p , and (cid:96) ∈ W p . Then, the followingasymptotic properties hold for (cid:101) β N .(a) (cid:101) β N is weakly consistent, uniformly in θ ∈ B . That is, for any ε > , lim N →∞ sup θ ∈ B P (cid:16)(cid:12)(cid:12) (cid:101) β N − θ (cid:12)(cid:12) > ε (cid:17) = 0 , (b) (cid:101) β N is asymptotically normal with parameter ( θ , I − N ) , uniformly in θ ∈ B , where I N is givenby (2.7) . That is, uniformly in θ ∈ B , (cid:112) I N (cid:0) (cid:101) β N − θ (cid:1) D −→ N (0 , , N → ∞ . (c) For any ε > , lim N →∞ sup θ ∈ B P (cid:16)(cid:12)(cid:12)(cid:12)(cid:112) I N (cid:0) (cid:101) β N − θ (cid:1) − ξ N (cid:12)(cid:12)(cid:12) > ε (cid:17) = 0 , where ξ N is defined as in (2.8) .(d) For any (cid:36) ∈ W p , (cid:101) β N is asymptotically efficient in B for the loss function (cid:36) N ( x ) := (cid:36) ( √ I N x ) . That is, for any θ ∈ B , lim δ → lim inf N →∞ sup | θ − θ | <δ E (cid:16) (cid:36) N (cid:0) (cid:101) β N − θ (cid:1)(cid:17) = E ( (cid:36) ( ξ )) , where ξ D = N (0 , under P .Proof. It is sufficient to check the conditions (N1) − (N4) in [IK81, Section III.1], which has beenverified in the proof of Theorem 3.2.With the help of the Bernstein-Von Mises theorem from next section, we will be able to inves-tigate the asymptotic properties of both (cid:101) β N and (cid:98) β N with a set of priors and loss functions havingexponential growth rates, as shown in Sections 4.2 and 4.3 below, respectively.2 Cheng, Cialenco, and Gong
We recall that, generally speaking, the Bernstein-von Mises type theorem states that the posteriordistribution of the normalized distance between the randomized parameter Θ and (cid:98) θ N (given as in(3.1)) is asymptotically normal. This type of result implies that the posterior distribution measureapproaches the Dirac measure as the number of observations increases; see Remark 4.5 belowfor more details. Moreover, it also serves as an essential tool in derivation of some asymptoticproperties of Bayesian estimators. To develop the Bernstein-Von Mises theorem regarding theposterior density, we adopt the techniques from [BKPR71] (see also [PR00] and [Bis02]), where weslightly weaken one of the conditions compared to some previous versions of Bernstein-Von Misestheorem (see condition (C2) in Theorem 4.4), which is also easier to verify.Let Λ := √ I N (Θ − (cid:98) θ N ), representing the normalized difference between the randomized param-eter Θ and (cid:98) θ N . By (4.2), the corresponding posterior density is then given by (cid:101) p ( λ | U N ) = p (cid:18) λ √ I N + (cid:98) θ N (cid:12)(cid:12)(cid:12)(cid:12) U N (cid:19) d θ d λ = C − N (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) e − λ / , λ ∈ ( − (cid:112) I N (cid:98) θ N , ∞ ) , (4.5)where C N := (cid:112) I N (cid:90) ∞ exp (cid:18) − I N (cid:0) η − (cid:98) θ N (cid:1) (cid:19) (cid:37) ( η ) d η = (cid:90) ∞−√ I N (cid:98) θ N (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) e − λ / d λ. (4.6)For notational convenience, we will extend the domains of (cid:37) and (cid:101) p to R , with (cid:37) ( θ ) = 0 for θ ∈ ( −∞ ,
0] and (cid:101) p ( λ | U N ) = 0 for λ ∈ ( −∞ , −√ I N (cid:98) θ N ]. By (4.5), the definition (4.4) of (cid:101) β N can bewritten as (cid:101) β N = arg min β ∈ R + (cid:90) R (cid:96) (cid:16) λ + (cid:112) I N (cid:0)(cid:98) θ N − β (cid:1)(cid:17) (cid:101) p ( λ | U N ) d λ. (4.7)We are now in the position of presenting the Bernstein-von Mises theorem. Theorem 4.4.
Let (cid:37) ∈ Q be positive and continuous in a neighborhood of θ , and let f be anon-negative, Borel-measurable function on R . Suppose that (cid:37) and f satisfy the following twoconditions:(C1) there exists α ∈ (0 , so that (cid:90) R f ( x ) e − αx / d x < ∞ ; (C2) for any δ > , lim N →∞ (cid:90) {| λ | > √ I N δ } f ( λ ) (cid:37) (cid:18)(cid:98) θ N + λ √ I N (cid:19) e − λ / d λ = 0 , P − a.s. . Then, lim N →∞ (cid:90) R f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ = 0 , P − a.s. . (4.8) ayesian Estimations for SPDEs Remark . The above theorem implies that, for P − a.s. ω , the posterior distribution measure of Θconverges weakly to the Dirac measure centered at θ , as the number of Fourier modes increases.Indeed, let g be any continuous and bounded function on R . Without loss of generality, assumethat g is non-negative (otherwise, consider g + and g − separately). By (4.5), for any ω ∈ Ω, (cid:90) R g ( θ ) p ( θ | U N )( ω ) d θ = (cid:90) R g (cid:18) λ √ I N + (cid:98) θ N ( ω ) (cid:19) (cid:101) p ( λ | U N )( ω ) d λ. For each given ω ∈ Ω, we put f ω ( λ ) := g ( λ/ √ I N + (cid:98) θ N ( ω )), λ ∈ R . In light of the boundedness of g , f ω satisfies conditions (C1) and (C2) above, for any given ω . From the proof of Theorem 4.4(see (4.15) below), equality (4.8) holds for f ω at any ω . Together with the strong consistency of (cid:98) θ N (recalling (3.6)), for P − a.s. ω ∈ Ω, we deduce thatlim N →∞ (cid:90) R g ( θ ) p ( θ | U N )( ω ) d θ = lim N →∞ (cid:90) R g (cid:18) λ √ I N + (cid:98) θ N ( ω ) (cid:19) (cid:101) p ( λ | U N )( ω ) d λ = lim N →∞ (cid:90) R f ω ( λ ) (cid:101) p ( λ | U N )( ω ) d λ = lim N →∞ √ π (cid:90) R f ω ( λ ) e − λ / d λ = lim N →∞ √ π (cid:90) R g (cid:18) λ √ I N + (cid:98) θ N ( ω ) (cid:19) e − λ / d λ = g ( θ ) , where we used the dominated convergence theorem in the last equality. Remark . Let (cid:37) ∈ Q , and let f be a non-negative, Borel-measurable function on R . Assumethat there exists c , c > r ∈ (0 ,
2) such that (cid:37) ( θ ) ≤ c e c θ r , for all θ ∈ R + ; f ( x ) ≤ c e c | x | r , for all x ∈ R . (4.9)Clearly such f satisfies condition (C1) in Theorem 4.4. Moreover, (cid:37) and f also satisfy condition(C2) in Theorem 4.4. Indeed, by the strong consistency of (cid:98) θ N , for P − a.s. ω ∈ Ω and N ∈ N largeenough (depending on ω ), we have that0 ≤ (cid:90) {| λ | > √ I N δ } f ( λ ) (cid:37) (cid:18)(cid:98) θ N + λ √ I N (cid:19) e − λ / d λ ≤ c (cid:90) {| λ | > √ I N δ } exp (cid:18) c | λ | r + c (cid:18) θ + 1 + | λ |√ I N (cid:19) r − λ (cid:19) { (cid:98) θ N + λ/ √ I N > } d λ ≤ c sup N ∈ N , λ ∈ R (cid:18) exp (cid:18)(cid:18) θ + 1 + | λ |√ I N (cid:19) r − λ (cid:19)(cid:19) · (cid:90) {| λ | > √ I N δ } exp (cid:18) c | λ | r − λ (cid:19) d λ → , (4.10)as N → ∞ . In particular, all conditions in Theorem 4.4 are valid when (cid:37) ∈ Q e, and f ∈ W e, .The proof of Theorem 4.4 is based on the following technical lemma. Lemma 4.7.
Under the conditions of Theorem 4.4,(a) there exist δ > such that lim N →∞ (cid:90) {| λ |≤ δ √ I N } f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ = 0 , P − a.s. ;4 Cheng, Cialenco, and Gong (b) for any δ > , lim N →∞ (cid:90) {| λ | >δ √ I N } f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ = 0 , P − a.s. . Proof.
Pick δ ∈ (0 , θ /
2) such that (cid:37) is continuous on [ θ − δ , θ + 2 δ ]. Recall that the Fisherinformation I N is unbounded, and thus, for any C >
0, there exists N ∈ N so that δ √ I N > C .We decompose the integral from part (a) as follows (cid:90) {| λ |≤ δ √ I N } = {| (cid:98) θ N − θ | >δ } (cid:90) {| λ |≤ δ √ I N } + {| (cid:98) θ N − θ |≤ δ } (cid:90) {| λ |≤ C } + {| (cid:98) θ N − θ |≤ δ } (cid:90) { C< | λ |≤ δ √ I N } . (4.11)By the strong consistency of (cid:98) θ N , | (cid:98) θ N ( ω ) − θ | ≤ δ , P − a.s., for sufficiently large N (that maydepend on ω ). Hencelim N →∞ {| (cid:98) θ N − θ | >δ } (cid:90) {| λ |≤ δ √ I N } f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ = 0 , P − a.s. . (4.12)Moreover, by the strong consistency of (cid:98) θ N and the continuity of (cid:37) on [ θ − δ , θ + 2 δ ], for any λ ∈ [ − C, C ], lim N →∞ (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , P − a.s. . Hence, by condition (C1) and the dominated convergence theorem,lim N →∞ {| (cid:98) θ N − θ |≤ δ } (cid:90) {| λ |≤ C } f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ = 0 , P − a.s. . (4.13)Finally, by (C1) and the boundedness of (cid:37) on [ θ − δ , θ + 2 δ ], {| (cid:98) θ N − θ |≤ δ } (cid:90) C< | λ |≤ δ √ I N f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ ≤ M (cid:90) {| λ | >C } f ( λ ) e − αλ / e − (1 − α ) λ / d λ ≤ M e − (1 − α ) C / (cid:90) R f ( λ ) e − αλ / dλ, (4.14)where M := sup θ ∈ [ θ − δ ,θ +2 δ ] (cid:37) ( θ ). Combining (4.11) − (4.14), and since C > δ > (cid:90) {| λ | >δ √ I N } f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ ≤ (cid:90) {| λ | >δ √ I N } f ( λ ) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) e − λ / d λ + (cid:37) ( θ ) e − (1 − α ) δ I N (cid:90) {| λ | >δ √ I N } f ( λ ) e − αλ / d λ, and in view of (C1) and (C2), (b) follows at once. The proof is now complete. ayesian Estimations for SPDEs Proof of Theorem 4.4.
Since the constant function f ≡ N →∞ (cid:90) R e − λ / (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) d λ = 0 , P − a.s. , which, together with (4.6), implies thatlim N →∞ C N := lim N →∞ (cid:90) R (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) e − λ / d λ = (cid:37) ( θ ) (cid:90) R e − λ / d λ = √ π(cid:37) ( θ ) , P − a.s. . Therefore, by (4.5), as N → ∞ , (cid:90) R f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ ≤ C − N (cid:90) R f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ + (cid:12)(cid:12)(cid:12)(cid:12) C − N (cid:37) ( θ ) − √ π (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R f ( λ ) e − λ / d λ → , P − a.s. , (4.15)which completes the proof of the theorem. (cid:3) In order to investigate the uniform asymptotic properties of (cid:101) β N , we need the following uniformversion of Bernstein-Von Mises Theorem. Theorem 4.8.
Let (cid:37) ∈ Q be continuous and positive on R + , and let f be a non-negative, Borel-measurable function on R . Assume that (cid:37) and f satisfy (4.9) . Then, for any compact set B ⊂ R + and any ε > , lim N →∞ sup θ ∈ B P (cid:18) (cid:90) R f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ > ε (cid:19) = 0 . The following technical lemma is a key ingredient for the proof of Theorem 4.8.
Lemma 4.9.
Under the conditions of Theorem 4.8, for any compact set B ⊂ R + ,(a) there exists δ > , such that for any ε > , lim N →∞ sup θ ∈ B P (cid:32)(cid:90) {| λ |≤ δ √ I N } f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ > ε (cid:33) = 0; (b) for any δ > and ε > , lim N →∞ sup θ ∈ B P (cid:32)(cid:90) {| λ | >δ √ I N } f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ > ε (cid:33) = 0 . Proof.
The proof follows the same line as that of Lemma 4.7. Pick δ ∈ (0 , inf θ ∈ B ( θ/ I N → + ∞ , for any C >
0, there exists N ∈ N so that δ √ I N > C . We decompose the integralfrom part (a) as in (4.11). By the uniform weak consistency of (cid:98) θ N (property (a) in Corollary 3.3),we have thatlim N →∞ sup θ ∈ B P (cid:32) {| (cid:98) θ N − θ | >δ } (cid:90) {| λ |≤ δ √ I N } f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ > ε (cid:33) ≤ lim N →∞ sup θ ∈ B P (cid:16)(cid:12)(cid:12)(cid:98) θ N − θ (cid:12)(cid:12) > δ (cid:17) = 0 . (4.16)6 Cheng, Cialenco, and Gong
Moreover, for any γ ∈ (0 , inf θ ∈ B θ ), let M γ := sup { θ ∈ B γ } (cid:37) ( θ ), where B γ := { θ ∈ R + : inf η ∈ B | η − θ | ≤ γ } is a compact subset of R + . Since C > θ ∈ B P (cid:32) {| (cid:98) θ N − θ |≤ δ } (cid:90) { C< | λ |≤ δ √ I N } f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ > ε (cid:33) ≤ sup θ ∈ B P (cid:18) M δ e − (1 − α ) C / (cid:90) R f ( λ ) e − αλ / dλ > ε (cid:19) = 0 . (4.17)Finally, since (cid:37) is uniformly continuous on B δ , there exists δ ∈ (0 , δ ) such that, for any θ, η ∈ B δ with | θ − η | ≤ δ , | (cid:37) ( θ ) − (cid:37) ( η ) | ≤ ε/ (6 CK ), where K := sup | λ |≤ C f ( λ ) < ∞ in view of (4.9). By theuniform weak consistency of (cid:98) θ N (property (a) in Corollary 3.3), for N ∈ N large enough (so that C/ √ I N ≤ δ / θ ∈ B P (cid:32) {| (cid:98) θ N − θ |≤ δ } (cid:90) {| λ |≤ C } f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ > ε (cid:33) ≤ sup θ ∈ B P (cid:32) sup | λ |≤ C (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) > ε CK , (cid:12)(cid:12)(cid:98) θ N − θ (cid:12)(cid:12) ≤ δ (cid:33) + sup θ ∈ B P (cid:18)(cid:12)(cid:12)(cid:98) θ N − θ (cid:12)(cid:12) > δ (cid:19) = sup θ ∈ B P (cid:18)(cid:12)(cid:12)(cid:98) θ N − θ (cid:12)(cid:12) > δ (cid:19) → , N → ∞ . (4.18)Combining (4.16) − (4.18) completes the proof of part (a).As for part (b), using similar arguments to those used in establishing (4.10), for any δ > δ ∈ (0 , inf θ ∈ B θ ), we have that, as N → ∞ ,sup θ ∈ B δ (cid:90) {| λ | >δ √ I N } f ( λ ) (cid:37) (cid:18) λ √ I N + θ (cid:19) e − λ / d λ ≤ c sup θ ∈ B δ ,λ ∈ R ,N ∈ N (cid:18) exp (cid:18)(cid:18) θ + λ √ I N (cid:19) r − λ (cid:19)(cid:19) · (cid:90) {| λ | > √ I N δ } exp (cid:18) c | λ | r − λ (cid:19) d λ → . This, together with the uniform weak consistency of (cid:98) θ N (property (a) in Corollary 3.3), impliesthat, for any ε > δ > θ ∈ B P (cid:32)(cid:90) {| λ | >δ √ I N } f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ > ε (cid:33) ≤ sup θ ∈ B P (cid:32)(cid:90) {| λ | >δ √ I N } f ( λ ) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) e − λ / d λ > ε (cid:33) + P (cid:32) sup θ ∈ B (cid:37) ( θ ) · (cid:90) {| λ | >δ √ I N } f ( λ ) e − λ / d λ > ε (cid:33) ≤ sup θ ∈ B P (cid:32)(cid:90) {| λ | >δ √ I N } f ( λ ) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) e − λ / d λ > ε , (cid:98) θ N ∈ B δ (cid:33) + sup θ ∈ B P (cid:0)(cid:98) θ N / ∈ B δ (cid:1) ≤ P (cid:32) sup θ ∈ B δ (cid:90) {| λ | >δ √ I N } f ( λ ) (cid:37) (cid:18) λ √ I N + θ (cid:19) e − λ / d λ > ε (cid:33) + sup θ ∈ B P (cid:16)(cid:12)(cid:12)(cid:98) θ N − θ (cid:12)(cid:12) > δ (cid:17) → , as N → ∞ , which completes the proof of part (b). ayesian Estimations for SPDEs Proof of Theorem 4.8.
Since B ⊂ R + is compact and ρ is positive on R + , L := inf θ ∈ B (cid:37) ( θ ) > ε >
0, by (4.5) and (4.6), we have thatsup θ ∈ B P (cid:32)(cid:90) R f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ > ε (cid:33) ≤ sup θ ∈ B P (cid:18) C − N (cid:90) R f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ > ε (cid:19) + sup θ ∈ B P (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) C − N (cid:37) ( θ ) − √ π (cid:12)(cid:12)(cid:12)(cid:12) (cid:90) R f ( λ ) e − λ / d λ > ε (cid:19) ≤ sup θ ∈ B P (cid:18)(cid:90) R f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ > ε C N , C N ≥ L (cid:19) + 2 sup θ ∈ B (cid:18) C N < L (cid:19) + sup θ ∈ B P (cid:32)(cid:12)(cid:12) √ π(cid:37) ( θ ) − C N (cid:12)(cid:12) (cid:90) R f ( λ ) e − λ / d λ > √ πε C N , C N ≥ L (cid:33) ≤ sup θ ∈ B P (cid:18)(cid:90) R f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ > Lε (cid:19) + 2 sup θ ∈ B (cid:18)(cid:12)(cid:12) C N − √ π(cid:37) ( θ ) (cid:12)(cid:12) > L (cid:19) + sup θ ∈ B P (cid:32)(cid:12)(cid:12) √ π(cid:37) ( θ ) − C N (cid:12)(cid:12) (cid:90) R f ( λ ) e − λ / d λ > √ πLε (cid:33) . (4.19)By Lemma 4.9, clearly we havelim N →∞ sup θ ∈ B P (cid:18)(cid:90) R f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ > Lε (cid:19) = 0 . (4.20)Moreover, for any δ >
0, by (4.6) and Lemma 4.9 (with f ≡
1) we have, as N → ∞ , thatsup θ ∈ B P (cid:16)(cid:12)(cid:12) C N − √ π(cid:37) ( θ ) (cid:12)(cid:12) > δ (cid:17) ≤ sup θ ∈ B P (cid:18)(cid:90) R (cid:12)(cid:12)(cid:12)(cid:12) (cid:37) (cid:18) λ √ I N + (cid:98) θ N (cid:19) − (cid:37) ( θ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / d λ > δ (cid:19) → . (4.21)Combining (4.19) − (4.21) completes the proof of the theorem. (cid:3) (cid:101) β N With the help of Theorems 4.4 and 4.8, we will now study the asymptotic properties of (cid:101) β N withrespect to the set of loss functions (cid:96) and prior densities (cid:37) which have exponential growth rates. Theorem 4.10.
Let (cid:37) ∈ Q e, and (cid:96) ∈ W e, . Assume that r (cid:55)→ (cid:82) R (cid:96) ( λ + r ) e − λ / d λ has a strictminimum at r = 0 , and that (cid:101) β N is well defined with respect to each loss function (cid:96) ∈ W e, , forevery N ∈ N . Then, lim N →∞ (cid:112) I N (cid:0) (cid:101) β N − (cid:98) θ N (cid:1) = 0 , P − a.s. . (4.22) Moreover, for any compact set B ⊂ R + and any ε > , lim N →∞ sup θ ∈ B P (cid:16)(cid:112) I N (cid:12)(cid:12) (cid:101) β N − (cid:98) θ N (cid:12)(cid:12) > ε (cid:17) = 0 . (4.23)8 Cheng, Cialenco, and Gong
Proof.
For any r ∈ R , denote by ψ ( r ) := (cid:90) R (cid:96) ( λ + r ) e − λ / √ π d λ, which has a strict minimum at r = 0. Note that the integral above is finite since (cid:96) ∈ W e, . Forany ε > N ∈ N , we first have that P (cid:16)(cid:112) I N (cid:12)(cid:12) (cid:101) β N − (cid:98) θ N (cid:12)(cid:12) > ε (cid:17) ≤ P (cid:18) inf β : √ I N | β − (cid:98) θ N | >ε (cid:90) R + (cid:96) (cid:0)(cid:112) I N ( η − β ) (cid:1) p ( η | U N ) d η < (cid:90) R + (cid:96) (cid:16)(cid:112) I N (cid:0) η − (cid:98) θ N (cid:1)(cid:17) p ( η | U N ) d η (cid:19) = P (cid:18) inf | r | >ε (cid:90) R (cid:96) ( λ + r ) (cid:101) p ( λ | U N ) d λ < (cid:90) R (cid:96) ( λ ) (cid:101) p ( λ | U N ) d λ (cid:19) ≤ P (cid:32) inf | r | >ε (cid:90) R (cid:96) ( λ + r ) (cid:101) p ( λ | U N ) d λ < ψ (0) + (cid:90) R (cid:96) ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ (cid:33) . (4.24)Pick δ = δ ( ε ) ∈ (0 , inf | r | >ε ψ ( r ) − ψ (0)). Since (cid:96) is symmetric on R and is non-decreasing on R + ,there exists m = m ( ε, δ ) > ε and M = M ( ε, δ ) > m such that (cid:90) m − m e − λ / √ π d λ ≥ inf | r | >ε ψ ( r ) − δ/ | r | >ε ψ ( r ) − δ/ , inf | x | >M − m (cid:96) ( x ) ≥ inf | r | >ε ψ ( r ) − δ . (4.25)To obtain the existence of M above, assume that for any x ∈ R , (cid:96) ( x ) ≤ inf | r | >ε ψ ( r ) − δ/
4, theninf | r | >ε ψ ( r ) = inf | r | >ε (cid:90) R (cid:96) ( λ + r ) e − λ / √ π d λ ≤ inf | r | >ε ψ ( r ) − δ , which is clearly a contradiction. Next, there exists K = K ( ε, δ ) > M large enough such that,whenever | r | ∈ ( ε, M ), (cid:90) K − K (cid:96) ( λ + r ) e − λ / √ π d λ ≥ (cid:90) R (cid:96) ( λ + r ) e − λ / √ π d λ − δ ≥ inf | r | >ε ψ ( r ) − δ . (4.26)Note that for | r | ≥ M , (4.25) implies that (cid:90) m − m (cid:96) ( λ + r ) e − λ / √ π d λ ≥ inf | r | >ε ψ ( r ) − δ/ | r | >ε ψ ( r ) − δ/ · (cid:18) inf | r | >ε ψ ( r ) − δ (cid:19) = inf | r | >ε ψ ( r ) − δ . (4.27)Hence, by combining (4.26) and (4.27), we obtain thatinf | r | >ε (cid:90) K − K (cid:96) ( λ + r ) e − λ / √ π d λ ≥ inf | r |≥ ε ψ ( r ) − δ > ψ (0) + δ . (4.28)Moreover, since (cid:96) is symmetric on R and is non-decreasing on R + , for any | r | > ε + 1 + 2 K , (cid:90) K − K (cid:96) ( λ + r ) (cid:101) p ( λ | U N ) d λ ≥ sup | x |≤ ε +1+ K (cid:96) ( x ) · (cid:90) K − K (cid:101) p ( λ | U N ) d λ ≥ (cid:90) K − K (cid:96) ( λ + ε + 1) (cid:101) p ( λ | U N ) d λ. ayesian Estimations for SPDEs | r | >ε (cid:90) R (cid:96) ( λ + r ) (cid:101) p ( λ | U N )d λ ≥ inf | r | >ε (cid:90) K − K (cid:96) ( λ + r ) (cid:101) p ( λ | U N )d λ = inf | r |∈ ( ε,ε +1+2 K ) (cid:90) K − K (cid:96) ( λ + r ) (cid:101) p ( λ | U N )d λ. Together with (4.28), we deduce thatinf | r | >ε (cid:90) R (cid:96) ( λ + r ) (cid:101) p ( λ | U N ) d λ ≥ inf | r |∈ ( ε,ε +1+2 K ) (cid:90) K − K (cid:96) ( λ + r ) e − λ / √ π d λ − sup | r |∈ ( ε,ε +1+2 K ) (cid:90) K − K (cid:96) ( λ + r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ> ψ (0) + δ − sup | x |≤ ε +1+3 K (cid:96) ( x ) · (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ. (4.29)Finally, by combining (4.24) and (4.29), we obtain that for any N ∈ N , P (cid:16)(cid:112) I N (cid:12)(cid:12) (cid:101) β N − (cid:98) θ N (cid:12)(cid:12) > ε (cid:17) ≤ P (cid:32)(cid:90) R (cid:18) (cid:96) ( λ ) + sup | x |≤ ε +1+3 K (cid:96) ( x ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ > δ (cid:33) , and thus P (cid:16) lim sup N →∞ (cid:110)(cid:112) I N (cid:12)(cid:12) (cid:101) β N − (cid:98) θ N (cid:12)(cid:12) > ε (cid:111)(cid:17) ≤ P (cid:32) lim sup N →∞ (cid:26) (cid:90) R (cid:0) (cid:96) ( λ ) + C (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ > δ (cid:27)(cid:33) , where C = C ( ε, δ, K ) := sup | x |≤ ε +1+3 K (cid:96) ( x ). The results of the theorem follow immediately fromTheorem 4.4 and 4.8. Remark . Theorem 4.10 remains valid when the symmetry of (cid:96) in condition (i) of Definition 3.1is relaxed to the non-increasing monotonicity on ( −∞ , (cid:82) R (cid:96) ( λ + r ) (cid:101) p ( λ | U N ) d λ into the integrals on [0 , ∞ ) and ( −∞ , Proposition 4.12.
Under the conditions of Theorem 4.10, the asymptotic properties (a) − (c) statedin Theorem 4.3 remain true for (cid:101) β N . Moreover, (cid:101) β N is strongly consistent. If in addition, (cid:37) ( x ) = R + ( x ) , is the uninformed/uniform prior, then (cid:101) β N also satisfies the asymptotic property ( d ) inTheorem 4.3.Proof. Properties (a) − (c) in Theorem 4.3 follow immediately from the properties (a) − (c) in Propo-sition 3.3 for (cid:98) θ N and Theorem 4.10. The strong consistency is a direct consequence of (3.6) and(4.22).Next, with (cid:96) ∈ W e, and satisfying the conditions of Theorem 4.10, we take (cid:37) ( x ) = R + ( x ).Due to (4.5) and (4.7), we obtain (cid:101) β N = arg min β ∈ R + (cid:90) ∞−√ I N (cid:98) θ N (cid:96) (cid:16) λ + (cid:112) I N (cid:0)(cid:98) θ N − β (cid:1)(cid:17) e − λ / d λ, (4.30)We will investigate the asymptotic efficiency of (cid:101) β N as stated in Theorem 4.3.(d). We will excludeany bounded loss function (cid:96) which clearly belong to W p , and thus their asymptotic efficiency iscovered by Theorem 4.3.0 Cheng, Cialenco, and Gong
In view of the asymptotic normality of (cid:101) β N with parameter ( θ , I − N ) (property (b) above),uniformly in θ ∈ B , and by [IK81, Theorem III.1.3], it is sufficient to show that for any m ∈ N ,there exists N = N ( m ) ∈ N , such that the family of random variables {|√ I N ( (cid:101) β N − θ ) | m , N ≥ N , θ ∈ B } are uniformly integrable under P , for any given compact set B ⊂ R + . In what follows,we will show that for any m ∈ N and θ ∈ B , there exists N = N ( m ) ∈ N and K = K ( m ) > N ≥ N and ω ∈ Ω, (cid:12)(cid:12)(cid:12)(cid:112) I N (cid:0) (cid:101) β N ( ω ) − θ (cid:1)(cid:12)(cid:12)(cid:12) m ≤ (cid:16)(cid:12)(cid:12)(cid:12)(cid:112) I N (cid:0)(cid:98) θ N ( ω ) − θ (cid:1)(cid:12)(cid:12)(cid:12) m + K (cid:17) , (4.31)which clearly implies the uniform integrability in light of (3.7).Since (cid:96) is symmetric on R and non-decreasing on R + , there exists x > (cid:96) ( x ) > (cid:96) ( x ) ≥ (cid:96) ( x − x ) for all x ≥ x , and (cid:96) ( x ) < (cid:96) ( x − x ) for all x ∈ [0 , x ). For such x , we can find x ∈ (0 , x ) so that (cid:96) ( x ) < (cid:96) ( x ). Then, for any N ∈ N and β ≥ x / √ I N , (cid:90) ∞−√ I N (cid:98) θ N (cid:18) (cid:96) (cid:16) λ + (cid:112) I N (cid:0)(cid:98) θ N − β (cid:1)(cid:17) − (cid:96) (cid:0) λ + (cid:112) I N (cid:98) θ N (cid:1)(cid:19) e − λ / d λ = (cid:90) R + (cid:16) (cid:96) (cid:0) y − (cid:112) I N β (cid:1) − (cid:96) ( y ) (cid:17) e − ( y −√ I N (cid:98) θ N ) / d y = (cid:90) x (cid:16) (cid:96) (cid:0) y − (cid:112) I N β (cid:1) − (cid:96) ( y ) (cid:17) e − ( y −√ I N (cid:98) θ N ) / d y + (cid:90) ∞ x (cid:16) (cid:96) (cid:0) y − (cid:112) I N β (cid:1) − (cid:96) ( y ) (cid:17) e − ( y −√ I N (cid:98) θ N ) / d y ≥ (cid:90) x (cid:16) (cid:96) (cid:0) y − (cid:112) I N β (cid:1) − (cid:96) ( y ) (cid:17) e − ( y −√ I N (cid:98) θ N ) / d y − (cid:90) ∞ x (cid:96) ( y ) e − ( y −√ I N (cid:98) θ N ) / d y ≥ (cid:0) (cid:96) (cid:0) x − x (cid:1) − (cid:96) ( x ) (cid:1) (cid:90) x e − ( y −√ I N (cid:98) θ N ) / d y − (cid:90) ∞ x (cid:96) ( y ) e − ( y −√ I N (cid:98) θ N ) / d y ≥ (cid:0) (cid:96) ( x ) − (cid:96) ( x ) (cid:1) (cid:90) x e − ( y −√ I N (cid:98) θ N ) / d y − (cid:90) R + (cid:96) ( z + x ) e − ( z + x −√ I N (cid:98) θ N ) / d z. Hence, for any N ∈ N , β ≥ x / √ I N , and ω ∈ Ω so that (cid:98) θ N ( ω ) < (cid:90) ∞−√ I N (cid:98) θ N ( ω ) (cid:18) (cid:96) (cid:16) λ + (cid:112) I N (cid:0)(cid:98) θ N ( ω ) − β (cid:1)(cid:17) − (cid:96) (cid:0) λ + (cid:112) I N (cid:98) θ N ( ω ) (cid:1)(cid:19) e − λ / d λ ≥ (cid:0) (cid:96) ( x ) − (cid:96) ( x ) (cid:1) x e − ( x −√ I N (cid:98) θ N ( ω )) / − e − ( x −√ I N (cid:98) θ N ( ω )) / (cid:90) R + (cid:96) ( z + x ) e − z / d z. Since x > x >
0, this implies that there exists K = K ( x , x ) > ω ∈ Ωwith √ I N (cid:98) θ N ( ω ) ≤ − K , the right-hand side of the above inequality is strictly positive. Hence, forany N ∈ N and β ≥ x / √ I N , we obtain that {√ I N (cid:98) θ N ≤− K } (cid:90) ∞−√ I N (cid:98) θ N (cid:96) (cid:16) λ + (cid:112) I N (cid:0)(cid:98) θ N − β (cid:1)(cid:17) e − λ / d λ> {√ I N (cid:98) θ N ≤− K } (cid:90) ∞−√ I N (cid:98) θ N (cid:96) (cid:0) λ + (cid:112) I N (cid:98) θ N (cid:1) e − λ / d λ, which, together with (4.30), implies that {√ I N (cid:98) θ N ≤− K } (cid:101) β N ≤ {√ I N (cid:98) θ N ≤− K } x √ I N , for any N ∈ N . ayesian Estimations for SPDEs (cid:101) β N ≥
0, we conclude that there exists N = N ( x ) ∈ N so that 2 x / (cid:112) I N ≤ inf θ ∈ B θ , andthat, for any θ ∈ B and N ≥ N , {√ I N (cid:98) θ N ≤− K } (cid:112) I N (cid:12)(cid:12) (cid:101) β N − θ (cid:12)(cid:12) ≤ {√ I N (cid:98) θ N ≤− K } (cid:112) I N θ ≤ {√ I N (cid:98) θ N ≤− K } (cid:112) I N (cid:12)(cid:12)(cid:98) θ N − θ (cid:12)(cid:12) . (4.32)It remains to prove (4.31), when √ I N (cid:98) θ N > − K . Since (cid:96) ∈ W e, is unbounded, there exists (cid:101) K > K large enough such that (cid:90) (cid:101) K K (cid:16) (cid:96) (cid:0) (cid:101) K (cid:1) − (cid:96) ( λ ) (cid:17) e − λ / d λ > (cid:90) [ − (cid:101) K , (cid:101) K ] c (cid:96) ( λ ) e − λ / d λ. Recalling that (cid:96) is symmetric, for any r < − (cid:101) K and N ∈ N , we deduce that {√ I N (cid:98) θ N ≥− K } (cid:90) ∞−√ I N (cid:98) θ N (cid:96) ( λ + r ) e − λ / d λ ≥ {√ I N (cid:98) θ N ≥− K } (cid:90) (cid:101) K ( −√ I N (cid:98) θ N ) ∨ ( − (cid:101) K ) (cid:96) ( λ + r ) e − λ / d λ ≥ {√ I N (cid:98) θ N ≥− K } (cid:96) (cid:0) (cid:101) K (cid:1) (cid:90) (cid:101) K ( −√ I N (cid:98) θ N ) ∨ ( − (cid:101) K ) e − λ / d λ> {√ I N (cid:98) θ N ≥− K } (cid:90) (cid:101) K ( −√ I N (cid:98) θ N ) ∨ ( − (cid:101) K ) (cid:96) ( λ ) e − λ / d λ + {√ I N (cid:98) θ N ≥− K } (cid:90) [ − (cid:101) K , (cid:101) K ] c (cid:96) ( λ ) e − λ / d λ> {√ I N (cid:98) θ N ≥− K } (cid:90) ∞−√ I N (cid:98) θ N (cid:96) ( λ ) e − λ / d λ, which, combined with (4.30), implies that {√ I N (cid:98) θ N ≥− K } (cid:112) I N (cid:0)(cid:98) θ N − (cid:101) β N (cid:1) ≥ {√ I N (cid:98) θ N ≥− K } (cid:0) − (cid:101) K (cid:1) , for any N ∈ N . (4.33)Using similar arguments as above, we can show that there exists K > K large enough, such that {√ I N (cid:98) θ N ≥− K } (cid:112) I N (cid:0)(cid:98) θ N − (cid:101) β N (cid:1) ≤ K {√ I N (cid:98) θ N ≥− K } , for any N ∈ N . (4.34)Finally, by combining (4.33) and (4.34) and letting K = max(2 (cid:101) K , K ), we obtain that, for any N ∈ N and θ ∈ B , {√ I N (cid:98) θ N ≥− K } (cid:112) I N (cid:12)(cid:12) (cid:101) β N − θ (cid:12)(cid:12) ≤ {√ I N (cid:98) θ N ≥− K } (cid:16)(cid:112) I N (cid:12)(cid:12)(cid:98) θ N − θ (cid:12)(cid:12) + K (cid:17) . (4.35)Combining (4.32) and (4.35) clearly leads to (4.31). The proof is now complete. Remark . The symmetry of the loss function (cid:96) is again not essential for the validity of Proposi-tion 4.12, and can be relaxed to the non-increasing monotonicity on ( −∞ ,
0) with a more technicalproof. Moreover, we also conjecture that the asymptotic efficiency for (cid:101) β N remains valid with anyprior (cid:37) ∈ Q e, , and the detail proof will be given elsewhere.Finally, while we have extended the choice of loss function (cid:96) to the class W e, with exponentialgrowth in the definition of (cid:101) β N for Theorem 4.10 and Proposition 4.12, we still keep (cid:36) ∈ W p in thedefinition of asymptotic efficiency for Corollary 4.12. The validity of the asymptotic efficiency for (cid:101) β N with more general (cid:36) ∈ W e, is yet to be studied.2 Cheng, Cialenco, and Gong (cid:98) β N In this section we will study the asymptotic properties of the Bayesian estimator (cid:98) β N . While we willinvestigate only (strong) consistency, and asymptotic normality of (cid:98) β N , we recall that (cid:98) β N is definedfor larger class of loss functions W (cid:48) , in comparison to (cid:101) β N from previous section.When (cid:98) β N is well defined for all N ∈ N , the following theorem provides a sufficient condition forthe consistency and asymptotic normality of (cid:98) β N . Theorem 4.14.
Assume that the prior density (cid:37) ∈ Q is positive and continuous in a neighborhoodof θ , and that (cid:98) β N is well-defined with respect to a loss function (cid:96) ∈ W (cid:48) , for every N ∈ N . Moreover,assume that there exists { a N } N ∈ N ⊂ R + , a test function f as in Theorem 4.4 satisfying conditions(C1) and (C2), and another loss function (cid:101) (cid:96) , such that(i) a N (cid:96) (cid:18) λ √ I N (cid:19) ≤ f ( λ ) , for any λ ∈ R and N ∈ N ;(ii) lim N →∞ sup λ ∈ B (cid:12)(cid:12)(cid:12)(cid:12) a N (cid:96) (cid:18) λ √ I N (cid:19) − (cid:101) (cid:96) ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , for any compact set B ⊂ R ;(iii) r (cid:55)→ (cid:90) R (cid:101) (cid:96) ( λ + r ) e − λ / d λ has a strict minimum at r = 0 .Then,(a) lim N →∞ (cid:112) I N (cid:0) (cid:98) β N − (cid:98) θ N (cid:1) = 0 , P − a.s.;(b) lim N →∞ a N (cid:90) R + (cid:96) (cid:0) θ − (cid:98) β N (cid:1) p ( θ | U N ) d θ = (cid:90) R (cid:101) (cid:96) ( λ ) e − λ / √ π d λ , P − a.s..In particular, (cid:98) β N is strongly consistent and asymptotically normal, as N → ∞ , namely (cid:98) β N → θ , P − a.s. and (cid:112) I N (cid:0) (cid:98) β N − θ (cid:1) D −→ N (0 , . Proof.
The strong consistency and asymptotic normality of (cid:98) β N are immediate consequences of part(a) together with the strong consistency and asymptotic normality of (cid:98) θ N (recalling (3.6) and (3.7)).The proof of (a) and (b) is split in four steps. Step 1.
We will first show thatlim sup N →∞ a N (cid:90) R + (cid:96) (cid:0) θ − (cid:98) β N (cid:1) p ( θ | U N ) d θ ≤ (cid:90) R (cid:101) (cid:96) ( λ ) e − λ / √ π d λ, P − a.s. . (4.36)By the definition of (cid:98) β N , (cid:90) R + (cid:96) (cid:0) θ − (cid:98) β N (cid:1) p ( θ | U N ) d θ ≤ (cid:90) R + (cid:96) (cid:0) θ − (cid:98) θ N (cid:1) p ( θ | U N ) d θ = (cid:90) R (cid:96) (cid:18) λ √ I N (cid:19) (cid:101) p ( λ | U N ) d λ. Hence, to prove (4.36), it suffices to show thatlim N →∞ a N (cid:90) R (cid:96) (cid:18) λ √ I N (cid:19) (cid:101) p ( λ | U N ) d λ = (cid:90) R (cid:101) (cid:96) ( λ ) e − λ / √ π d λ, P − a.s. . (4.37) ayesian Estimations for SPDEs (cid:101) (cid:96) ( λ ) ≤ f ( λ ), for any λ ∈ R . Therefore, by conditions(i) and (ii), Theorem 4.4, and the dominated convergence theorem, we deduce thatlim N →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a N (cid:90) R (cid:96) (cid:18) λ √ I N (cid:19)(cid:101) p ( λ | U N ) d λ − (cid:90) R (cid:101) (cid:96) ( λ ) e − λ / √ π d λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim N →∞ (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12) a N (cid:96) (cid:18) λ √ I N (cid:19) − (cid:101) (cid:96) ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ + lim N →∞ (cid:90) R (cid:101) (cid:96) ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ + lim N →∞ (cid:90) R (cid:12)(cid:12)(cid:12)(cid:12) a N (cid:96) (cid:18) λ √ I N (cid:19) − (cid:101) (cid:96) ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) e − λ / √ π d λ ≤ lim N →∞ (cid:90) R f ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p ( λ | U N ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ + (cid:90) R (cid:18) lim N →∞ (cid:12)(cid:12)(cid:12)(cid:12) a N (cid:96) (cid:18) λ √ I N (cid:19) − (cid:101) (cid:96) ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) e − λ / √ π d λ ≤ K →∞ (cid:90) K − K (cid:18) lim N →∞ (cid:12)(cid:12)(cid:12)(cid:12) a N (cid:96) (cid:18) λ √ I N (cid:19) − (cid:101) (cid:96) ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) e − λ / √ π d λ ≤ lim K →∞ (cid:32) lim N →∞ sup λ ∈ [ − K,K ] (cid:12)(cid:12)(cid:12)(cid:12) a N (cid:96) (cid:18) λ √ I N (cid:19) − (cid:101) (cid:96) ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:33) (cid:90) K − K e − λ / √ π d λ = 0 , which completes the proof of (4.37), and therefore, of (4.36). Step 2.
Next, we will show that the sequence of random variables Y N := √ I N ( (cid:98) β N − (cid:98) θ N ), N ∈ N ,are uniformly bounded, P − a.s.. That is, for P − a.s. ω , there exists K ( ω ) ∈ (0 , ∞ ), such that | Y N ( ω ) | ≤ K ( ω ) for all N ∈ N .For any ω ∈ A := { ω ∈ Ω : lim sup N →∞ | Y N ( ω ) | = ∞} and any K ∈ N , there exists anincreasing sequence of integers N j = N j ( ω, K ), j ∈ N , such that N j ↑ ∞ , as j → ∞ , and | Y N j ( ω ) | ≥ K , for any j ∈ N . Consequently, (cid:90) R + (cid:96) (cid:0) θ − (cid:98) β N j ( ω ) (cid:1) p (cid:0) θ | U N j (cid:1) ( ω ) d θ = (cid:90) R (cid:96) (cid:18) λ + Y N j ( ω ) (cid:112) I N j (cid:19) (cid:101) p (cid:0) λ | U N j (cid:1) ( ω ) d λ ≥ (cid:90) K − K (cid:96) (cid:18) λ + K (cid:112) I N j (cid:19) (cid:101) p (cid:0) λ | U N j (cid:1) ( ω ) d λ, (4.38)where we used (3.4) in the last inequality. On the other hand, since (cid:101) (cid:96) is locally bounded, thefunction λ (cid:55)→ [ − K,K ] ( λ ) (cid:101) (cid:96) ( λ + K ) is bounded on R and thus satisfies conditions (C1) and (C2).Hence, by Theorem 4.4 and condition (ii), as j → ∞ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a N j (cid:90) K − K (cid:96) (cid:18) λ + K (cid:112) I N j (cid:19) (cid:101) p (cid:0) λ | U N j (cid:1) d λ − (cid:90) K − K (cid:101) (cid:96) ( λ + K ) e − λ / √ π d λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) K − K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a N j (cid:96) (cid:18) λ + K (cid:112) I N j (cid:19) − (cid:101) (cid:96) ( λ + K ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:101) p (cid:0) λ | U N j (cid:1) d λ + (cid:90) K − K (cid:101) (cid:96) ( λ + K ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p (cid:0) λ | U N j (cid:1) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ ≤ sup λ ∈ [ − K, K ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a N j (cid:96) (cid:18) λ (cid:112) I N j (cid:19) − (cid:101) (cid:96) ( λ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:90) K − K (cid:101) (cid:96) ( λ + K ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p (cid:0) λ | U N j (cid:1) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ → , P − a.s. . (4.39)Let B denote the exceptional subset of Ω in which the limit in (4.39) does not hold, then P ( B ) = 0.4 Cheng, Cialenco, and Gong
Without loss of generality, assume that A \ B (cid:54) = ∅ . By (4.38) and (4.39), for any ω ∈ A \ B ,lim sup j →∞ a N j (cid:90) R + (cid:96) (cid:0) θ − (cid:98) β N j ( ω ) (cid:1) p (cid:0) θ | U N j (cid:1) ( ω ) d θ ≥ lim j →∞ (cid:90) K − K (cid:96) (cid:18) λ + K (cid:112) I N j (cid:19) (cid:101) p (cid:0) λ | U N j (cid:1) ( ω ) d λ = (cid:90) K − K (cid:101) (cid:96) ( λ + K ) e − λ / √ π d λ. Since K ∈ N is arbitrary, by condition (iii) and monotone convergence theorem, we havelim sup j →∞ a N j (cid:90) R + (cid:96) (cid:0) θ − (cid:98) β N j ( ω ) (cid:1) p (cid:0) θ | U N j (cid:1) ( ω ) d θ ≥ lim K →∞ (cid:90) K − K (cid:101) (cid:96) ( λ + K ) e − λ / √ π dλ > (cid:90) R (cid:101) (cid:96) ( λ ) e − λ / √ π d λ. In view of (4.36), we must have P ( A \ B ) = 0 so that P ( A ) = 0, completing the proof of Step 2. Step 3.
We now prove (a) and (b). By Step 2, for P − a.s. ω , there exists N j = N j ( ω ) ∈ N , j ∈ N ,such that the sequence ( Y N j ( ω )) j ∈ N is convergent, as j → ∞ , and we denote its limit by Y ∗ ( ω ).For any K > a N j (cid:90) R + (cid:96) (cid:0) θ − (cid:98) β N j ( ω ) (cid:1) p (cid:0) θ | U N j (cid:1) ( ω ) d θ = a N j (cid:90) R (cid:96) (cid:18) λ + Y N j ( ω ) (cid:112) I N j (cid:19) (cid:101) p (cid:0) λ | U N j (cid:1) d λ ≥ a N j (cid:90) K − K (cid:96) (cid:18) λ + Y N j ( ω ) (cid:112) I N j (cid:19) (cid:101) p (cid:0) λ | U N j (cid:1) d λ. (4.40)Moreover, for j ∈ N large enough, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a N j (cid:90) K − K (cid:96) (cid:18) λ + Y N j ( ω ) (cid:112) I N j (cid:19) (cid:101) p (cid:0) λ | U N j (cid:1) d λ − (cid:90) K − K (cid:101) (cid:96) ( λ + Y ∗ ( ω )) e − λ / √ π d λ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:90) K − K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a N j (cid:96) (cid:18) λ + Y N j ( ω ) (cid:112) I N j (cid:19) − (cid:101) (cid:96) (cid:0) λ + Y N j ( ω ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:101) p (cid:0) λ | U N j (cid:1) ( ω ) d λ + (cid:90) K − K (cid:101) (cid:96) (cid:0) λ + Y N j ( ω ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p (cid:0) λ | U N j (cid:1) ( ω ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ + (cid:90) K − K (cid:12)(cid:12)(cid:12)(cid:101) (cid:96) (cid:0) λ + Y N j ( ω ) (cid:1) − (cid:101) (cid:96) ( λ + Y ∗ ( ω )) (cid:12)(cid:12)(cid:12) e − λ / √ π d λ ≤ (cid:90) K − K (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a N j (cid:96) (cid:18) λ + Y N j ( ω ) (cid:112) I N j (cid:19) − (cid:101) (cid:96) (cid:0) λ + Y N j ( ω ) (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:101) p (cid:0) λ | U N j (cid:1) ( ω ) d λ + (cid:90) K − K (cid:16) ˜ (cid:96) ( λ + Y ∗ ( ω ) + 1) [0 , ∞ ) ( λ ) + (cid:101) (cid:96) ( λ + Y ∗ ( ω ) − ( −∞ , ( λ ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:12)(cid:101) p (cid:0) λ | U N j (cid:1) ( ω ) − e − λ / √ π (cid:12)(cid:12)(cid:12)(cid:12) d λ + (cid:90) K − K (cid:12)(cid:12)(cid:12)(cid:101) (cid:96) (cid:0) λ + Y N j ( ω ) (cid:1) − (cid:101) (cid:96) ( λ + Y ∗ ( ω )) (cid:12)(cid:12)(cid:12) e − λ / √ π d λ. (4.41)By the same argument as in (4.39), the first two integrals in (4.41) vanish as j → ∞ , for P − a.s. ω .Moreover, the monotonicity property (3.4) for (cid:101) (cid:96) implies that it is almost surely (with respect tothe Lebesgue measure) continuous on R . Hence, conditions (i) and (ii) (which implies that (cid:101) (cid:96) isbounded by f ), together with the dominated convergence theorem, imply that the last integral in(4.41) vanishes as j → ∞ , for P − a.s. ω . Therefore, for P − a.s. ω , we havelim j →∞ a N j (cid:90) K − K (cid:96) (cid:18) λ + Y N j ( ω ) (cid:112) I N j (cid:19) (cid:101) p (cid:0) λ | U N j (cid:1) d λ = (cid:90) K − K (cid:101) (cid:96) ( λ + Y ∗ ( ω )) e − λ / √ π d λ. (4.42) ayesian Estimations for SPDEs K > j →∞ a N j (cid:90) R + (cid:96) (cid:0) θ − (cid:98) β N j (cid:1) p (cid:0) θ | U N j (cid:1) d θ ≥ (cid:90) R (cid:101) (cid:96) ( λ + Y ∗ ( ω )) e − λ / √ π d λ ≥ (cid:90) R (cid:101) (cid:96) ( λ ) e − λ / √ π d λ, P − a.s. , (4.43)where we have used condition (iii) in the second inequality.Now assume that P ( Y ∗ (cid:54) = 0) >
0. Then inequality on (4.43) would be strict in { Y ∗ (cid:54) = 0 } byassumption (iii), which is clearly a contradiction to (4.36). Therefore, P ( Y ∗ (cid:54) = 0) = 0. Combining(4.36) in Step 1 with (4.43) completes the proof of (b).To prove (a), it remains to show that every subsequence of ( Y N ) N ∈ N converges to 0 with proba-bility one. Indeed, assume the contrary. Applying similar arguments as in Step 3 to this exceptionalsubsequence leads to a contradiction to (4.36).The proof is now complete. Remark . Here we present an example of loss function in W e, that satisfies conditions (i) − (iii)in Theorem 4.14 above. Let (cid:96) ( x ) = exp( | x | r ) − r ∈ (0 , a N = I r/ N . Without loss ofgenerality, we assume I N ≥ N ∈ N . Note that a N (cid:96) (cid:18) λ √ I N (cid:19) = I r/ N (cid:16) e | λ | r I − r/ N − (cid:17) ≤ | λ | r e | λ | r I − r/ N ≤ | λ | r e | λ | r =: f ( λ ) , which satisfies conditions (C1) and (C2), due to Remark 4.6. Hence, condition (i) is valid. Moreover,let (cid:101) (cid:96) ( x ) = | x | r , which clearly satisfies condition (iii). For any K > λ ∈ [ − K, K ], (cid:12)(cid:12)(cid:12)(cid:12) a N (cid:96) (cid:18) λ √ I N (cid:19) − (cid:101) (cid:96) ( λ ) (cid:12)(cid:12)(cid:12)(cid:12) = I r/ N (cid:16) e | λ | r I − r/ N − − | λ | r I − r/ N (cid:17) ≤ K r I − r/ N e K r I − r/ N → , as N → ∞ , which shows the validity of condition (ii). In this section, we provide an illustrative numerical example of the asymptotics of the MLE andthe Bayesian estimators derived in the previous sections. Specifically, we consider the followingequation d u ( t, x ) + θ ( − u xx ( t, x )) d t = σ (cid:80) ∞ k =1 u k ( t ) h k ( x ) k α d w k ( t ) , t > u (0 , x ) = π − (cid:0) x − π (cid:1) , x ∈ [0 , π ] ,u ( t,
0) = u ( t, π ) = 0 , t > , (5.1)where h k ( x ) := (cid:112) /π sin( kx ), x ∈ [0 , π ], and α ∈ R . We will fix σ = 1 and T = 1 for the rest of thissection. We note that in this case, A = − ∂ xx , with its eigenfunctions h k , k ∈ N , and correspondingeigenvalues µ k = k , k ∈ N . In view of (2.7), the Fisher information I N ∝ N − α .We will consider two set of parameters, one for which condition (E1) from Remark 2.2 is satisfied,and another one that corresponds to condition (E2) from the same remark. All evaluations areperformed using numerical computing environment Matlab , and the source codes are availablefrom the authors upon request.6
Cheng, Cialenco, and Gong
Parameter Set I.
We take the true value of the parameter of interest to be θ = 0 .
3, and α < α ↑
1, the rate of divergence of I N as N → ∞ , and thus the rates ofconvergence of both the MLE and the Bayesian estimators, become smaller, as N → ∞ . We willillustrate this below by considering two different regimes α = 0 and α = 0 . N = 20 independent Brownian sample paths { w k ( t ) , t ∈ [0 , } , k = 1 , . . . ,
20, withtime-step of 5 × − , and compute the corresponding Fourier modes { u k ( t ) , t ∈ [0 , } , k = 1 , . . . , θ . Then, we compute (cid:98) θ N according to (3.1), where the Itˆo integral is approximatedby a finite sum with time-step of 5 × − . We also computed (cid:98) θ N using (3.2), that yielded sameresults and they will not be reported here. Moreover, we simulate the posterior density p ( θ | U N )using (4.2), where the integral in the denominator is approximated using the ‘integral’ Matlab built-in function.In Figures 1 and 2, we present and compare the posterior density p ( θ | U N ) under two differentprior distributions, the uniform uninformative prior on R + , (cid:37) ( x ) = R + ( x ) (left panels), andtruncated normal N (1 , . R + ) (right panels), for α = 0 and α = 0 . N = 2 (black square on the horizontal axis), and N = 4(black diamond on the horizontal axis). For both choices of α , the posterior densities under bothpriors converge to the Dirac measure concentrated at the MLEs, which is consistent with thediscussion in Remark 4.5. Moreover, under both priors, the posterior densities with α = 0 exhibitfaster convergence rates than those with α = 0 .
999 as expected.
Figure 1:
Parameter Set I. Posterior densities for N = 2 , , two different priors, and α = 0 . Next, we take the quadratic loss function (cid:96) ( x ) = x , in which case (cid:101) β N and (cid:98) β N are given by (cid:101) β N = (cid:98) β N = (cid:90) R + θ p ( θ | U N ) d θ. Also note that such Bayesian estimator can be viewed as the conditional mean estimator. InFigures 3 and 4, we compare (cid:98) θ N with the Bayesian estimator (cid:98) β N , using the two considered priorsand for α = 0 and α = 0 . (cid:98) θ N and the Bayesian estimators converge to the trueparameter θ = 0 .
3, as the number of Fourier modes increases, for both choices of α , the caseof α = 0 tends to have a better convergence rate. Finally, in Figure 5, we display the valuesof √ I N (cid:12)(cid:12)(cid:12) (cid:98) β N − (cid:98) θ N (cid:12)(cid:12)(cid:12) as function of N , that confirm the asymptotic results of Theorem 4.10 andTheorem 4.14. ayesian Estimations for SPDEs Figure 2:
Parameter Set I. Posterior densities for N = 2 , , , two different priors, and α = 0 . . Figure 3:
Parameter Set I. MLE vs. Bayesian Estimator with α = 0 . Figure 4:
Parameter Set I. MLE vs. Bayesian Estimator with α = 0 . . Parameter Set II.
In the second set of parameters, we let θ = 0 .
505 and α = 1, which correspondsto the case (E2) from Remark 2.2. We consider the same two priors as above. The posterior densities8 Cheng, Cialenco, and Gong
Figure 5:
Parameter Set I. √ I N (cid:12)(cid:12)(cid:12) (cid:98) β N − (cid:98) θ N (cid:12)(cid:12)(cid:12) , for two different priors, and α = 0 , and α = 0 . . behave similarly as in Case I. In contrast to Case I, we consider a loss function with exponentialgrowth, namely, (cid:96) ( x ) = exp (cid:0) | x | / (cid:1) −
1, and we compute both (cid:101) β N and (cid:98) β N using (cid:96) . In Figure 6 weplot the values of MLE and the two Bayesian estimators as function of N . All estimators performwell, and similarly. Although we display the results for one path, similar behavior is observed onother simulated paths of the solution. The values of scaled errors √ I N | (cid:101) β N − (cid:98) θ N | and √ I N | (cid:98) β N − (cid:98) θ N | are displayed in Figure 7, again confirming the results of Theorem 4.10 and Theorem 4.14. Figure 6:
Parameter Set II. Value of (cid:98) θ N , (cid:101) β N , (cid:98) β N , for two different priors, and α = 1 . A Auxiliary Results
For the sake of completeness, we recall a version of the strong law of large number, which will beused in the proof of the strong consistency of MLE. We refer the reader to [Shi96, Theorem IV.3.2]for its detail proof.
Theorem A.1 (Strong Law of Large Numbers) . Let { ξ n } n ∈ N be a sequence of independent randomvariables with finite second moments. Let { b n } n ∈ N be a sequence of non-decreasing real numbers ayesian Estimations for SPDEs Figure 7:
Parameter Set II. Scaled errors √ I N (cid:12)(cid:12)(cid:12) (cid:101) β N − (cid:98) θ N (cid:12)(cid:12)(cid:12) , √ I N (cid:12)(cid:12)(cid:12) (cid:98) β N − (cid:98) θ N (cid:12)(cid:12)(cid:12) , for two different priors,and α = 1 such that lim n →∞ b n = ∞ , and (cid:80) ∞ n =1 Var( ξ n ) /b n < ∞ . Then, (cid:80) nk =1 ( ξ k − E ( ξ k )) b n → , P − a.s. , n → ∞ . Also here we present a simple technical lemma used in Section 3.
Lemma A.2.
If the sequence { a n } n ∈ N ⊂ R satisfies a > and a k ≥ , k ≥ , then N (cid:88) n =1 a n ( (cid:80) nk =1 a k ) < + ∞ . Proof.
Note the following N (cid:88) n =1 a n ( (cid:80) nk =1 a k ) ≤ a + N (cid:88) n =2 a n ( (cid:80) n − k =1 a k )( (cid:80) nk =1 a k )= 1 a + N (cid:88) n =2 (cid:32) (cid:80) n − k =1 a k − (cid:80) nk =1 a k (cid:33) = 1 a + 1 a − (cid:80) Nk =1 a k , which finishes the proof. B Discussions on derivation of the posterior density
In this appendix, we will present a formal derivation of the posterior density (4.1). We make thefollowing standing assumptions(i) The random variable Θ is independent of the Brownian motions { w k , k ∈ N } and possesses aprobability density function (cid:37) ;(ii) σ (Θ) ⊂ F .0 Cheng, Cialenco, and Gong
We let θ stand for the dummy variable of Θ. Recall that U θN denotes the first N Fourier modesof the solution u θ to (2.1) with parameter θ . Let U Θ N be the R N -valued process obtained bysubstituting θ with Θ in U θN . By condition (i), for any B (( R N ) [0 ,T ] ) ⊗ B ( R + )-measurable functional f on ( R N ) [0 ,T ] × R + with E ( f ( U Θ N , Θ)) < ∞ , we have that E (cid:0) f (cid:0) U Θ N , Θ (cid:1)(cid:1) = (cid:90) R + E (cid:16) f (cid:16) U θN , θ (cid:17)(cid:17) (cid:37) ( θ ) d θ. (B.1)Next, we will show that, for any B ∈ B ( R + ), P (cid:0) Θ ∈ B | U Θ N (cid:1) = (cid:90) B r ηN (cid:0) U Θ N (cid:1) (cid:37) ( η ) d η (cid:90) R + r ηN (cid:0) U Θ N (cid:1) (cid:37) ( η ) d η , (B.2)where, r ηN (cid:16) U θN (cid:17) := exp (cid:32) − ησ N (cid:88) k =1 µ k q − k (cid:90) T d u θk ( t ) u θk ( t ) − η T σ N (cid:88) k =1 µ k q − k (cid:33) , for any θ, η ∈ R + For the sake of argument, we assume that r ηN ( · ) is B (( R N ) [0 ,T ] )-measurable.Clearly, (B.2) follows immediately from the following equality E (cid:18) { Θ ∈ B } (cid:90) R + r ηN (cid:0) U Θ N (cid:1) (cid:37) ( η ) d η (cid:12)(cid:12)(cid:12)(cid:12) U Θ N (cid:19) = (cid:90) B r ηN (cid:0) U Θ N (cid:1) (cid:37) ( η ) d η. (B.3)To prove (B.3), for any A ∈ B (( R N ) [0 ,T ] ), we first deduce from (B.1) that E (cid:18) { U Θ N ∈ A } { Θ ∈ B } (cid:90) R + r ηN (cid:0) U Θ N (cid:1) (cid:37) ( η ) d η (cid:19) = (cid:90) B E (cid:18) { U θN ∈ A } (cid:90) R + r ηN (cid:0) U θN (cid:1) (cid:37) ( η ) d η (cid:19) (cid:37) ( θ ) d θ = (cid:90) R + (cid:18)(cid:90) B E (cid:16) { U θN ∈ A } r ηN (cid:0) U θN (cid:1)(cid:17) (cid:37) ( θ ) d θ (cid:19) (cid:37) ( η ) d η. In view of (2.6), we have d P ηN d P θN (cid:16) U θN (cid:17) = r ηN (cid:0) U θN (cid:1) r θN (cid:0) U θN (cid:1) . (B.4)Hence, by Girsanov theorem, E (cid:18) { U Θ N ∈ A } { Θ ∈ B } (cid:90) R + r ηN (cid:0) U Θ N (cid:1) (cid:37) ( η )d η (cid:19) = (cid:90) R + (cid:18)(cid:90) B E (cid:18) d P ηN d P θN (cid:0) U θN (cid:1) { U θN ∈ A } r θN (cid:0) U θN (cid:1)(cid:19) (cid:37) ( θ )d θ (cid:19) (cid:37) ( η )d η = (cid:90) R + (cid:18)(cid:90) B E (cid:16) { U ηN ∈ A } r θN (cid:0) U ηN (cid:1)(cid:17) (cid:37) ( θ ) d θ (cid:19) (cid:37) ( η ) d η. (B.5)On the other hand, E (cid:18) { U Θ N ∈ A } (cid:90) B r ηN (cid:0) U Θ N (cid:1) (cid:37) ( η ) d η (cid:19) = (cid:90) R + E (cid:18) { U θN ∈ A } (cid:90) B r ηN (cid:0) U θN (cid:1) (cid:37) ( η ) d η (cid:19) (cid:37) ( θ ) d θ = (cid:90) R + (cid:18)(cid:90) B E (cid:16) { U θN ∈ A } r ηN (cid:0) U θN (cid:1)(cid:17) (cid:37) ( η ) d η (cid:19) (cid:37) ( θ ) d θ. (B.6)Combining (B.5) and (B.6) leads to (B.3). ayesian Estimations for SPDEs References [Bis99] J. P. N. Bishwal. Bayes and sequential estimation in Hilbert space valued stochasticdifferential equations.
J. Korean Statist. Soc. , 28(1):93–106, 1999.[Bis02] J. P. N. Bishwal. The Bernstein-von Mises theorem and spectral asymptotics of Bayesestimators for parabolic SPDEs.
J. Aust. Math. Soc. , 72(2):287–298, 2002.[BKPR71] J. Borwanker, G. Kallianpur, and B. L. S. Prakasa Rao. The Bernstein-von Misestheorem for Markov processes.
Ann. Math. Statist. , 42:1241–1253, 1971.[BT17] M. Bibinger and M. Trabs. Volatility estimation for stochastic PDEs using high-frequency observations.
Preprint, arXiv:1710.03519 , 2017.[CH17] I. Cialenco and Y. Huang. A note on parameter estimation for discretely sampledSPDEs.
Preprint, arXiv:1710.01649 , 2017.[Cho07] P. Chow.
Stochastic partial differential equations . Chapman & Hall/CRC AppliedMathematics and Nonlinear Science Series. Chapman & Hall/CRC, Boca Raton, FL,2007.[Cia10] I. Cialenco. Parameter estimation for SPDEs with multiplicative fractional noise.
Stoch. Dyn. , 10(4):561–576, 2010.[Cia18] I. Cialenco. Statistical inference for SPDEs: an overview.
Statistical Inference forStochastic Processes , 21(2):309–329, 2018.[CL09] I. Cialenco and S. V. Lototsky. Parameter estimation in diagonalizable bilinearstochastic parabolic equations.
Stat. Inference Stoch. Process. , 12(3):203–219, 2009.[FGHV16] S. Friedlander, N. Glatt-Holtz, and V. Vicol. Inviscid limits for a stochastically forcedshell model of turbulent flow.
Ann. Inst. Henri Poincar´e Probab. Stat. , 52(3):1217–1247, 2016.[GHKVZ14] N. Glatt-Holtz, I. Kukavica, V. Vicol, and M. Ziane. Existence and regularity ofinvariant measures for the three dimensional stochastic primitive equations.
J. Math.Phys. , 55(5):051504, 34, 2014.[GHZ08] N. Glatt-Holtz and M. Ziane. The stochastic primitive equations in two space dimen-sions with multiplicative noise.
Discrete Contin. Dyn. Syst. Ser. B , 10(4):801–822,2008.[GHZ09] N. Glatt-Holtz and M Ziane. Strong pathwise solutions of the stochastic Navier-Stokessystem.
Adv. Differential Equations , 14(5-6):567–600, 2009.[IK81] I. A. Ibragimov and R. Z. Khasminskii.
Statistical estimation , volume 16 of
Applica-tions of Mathematics . Springer-Verlag, New York, 1981.[LR17] S. V. Lototsky and B. L. Rozovsky.
Stochastic partial differential equations . Univer-sitext. Springer International Publishing, 2017.[LR18] S. V. Lototsky and B. L. Rozovsky.
Stochastic Evolution Systems. Linear theory andapplications to non-linear filtering , volume 89 of
Probability Theory and StochasticModelling . Springer International Publishing, 2018.2
Cheng, Cialenco, and Gong [LS00] R. S. Liptser and A. N. Shiryayev.
Statistics of random processes I. General theory .Springer-Verlag, New York, 2nd edition, 2000.[PR00] B. L. S. Prakasa Rao. Bayes estimation for some stochastic partial differentialequations.
J. Statist. Plann. Inference , 91(2):511–524, 2000. Prague Workshop onPerspectives in Modern Statistical Inference: Parametrics, Semi-parametrics, Non-parametrics (1998).[PT07] J. Posp´ıˇsil and R. Tribe. Parameter estimates and exact variations for stochastic heatequations driven by space-time white noise.
Stoch. Anal. Appl. , 25(3):593–611, 2007.[Shi96] A. N. Shiryaev.
Probability , volume 95 of