The Bernstein-von Mises theorem and spectral asymptotics of Bayes estimators for parabolic SPDEs

Abstract

The Bemstein-von Mises theorem, concerning the convergence of suitably normalized and centred posterior density to normal density, is proved for a certain class of linearly parametrized parabolic stochastic partial differential equations (SPDEs) as the number of Fourier coefficients in the expansion of the solution increases to infinity. As a consequence, the Bayes estimators of the drift parameter, for smooth loss functions and priors, are shown to be strongly consistent, asymptotically normal and locally asymptotically minimax (in the Hajek-Le Cam sense), and asymptotically equivalent to the maximum likelihood estimator as the number of Fourier coefficients become large. Unlike in the classical finite dimensional SDEs, here the total observation time and the intensity of noise remain fixed.2000 Mathematics subject classification: primary 6OH15,62M05, 62F12, 62F15.

Statistical inference for stochastic parabolic equations: a spectral approach

“…Alternatively, it follows from the definition of ξ(a) that 13) and, for each t > s, the random variables X(t; a), X(s; a) are jointly Gaussian with zero mean and correlation coefficient…”

“…(2) Bayesian estimators and hypothesis testing: Bishwal [12], [13], Prakasa Rao [59]. (3) Time-dependent drift: Huebner and Lototsky [27], [28].…”

Bayesian estimations for diagonalizable bilinear SPDEs

“…Moreover, it also serves as an essential tool in derivation of some asymptotic properties of Bayesian estimators. To develop the Bernstein-Von Mises theorem regarding the posterior density, we adopt the techniques from [BKPR71] (see also [PR00] and [Bis02]), where we slightly weaken one of the conditions compared to some previous versions of Bernstein-Von Mises theorem (see condition (C2) in Theorem 4.4), which is also easier to verify.…”

“…Asymptotic properties of the estimators are studied in the large number of Fourier modes regime, N → ∞, while time horizon is fixed. In particular, there are only few works related to Bayesian statistics for infinite dimensional evolution equations [Bis02,Bis99,PR00]. As usual, studying SPDEs driven by multiplicative noise is more involved, and the parameter estimation problems for such equations are not an exception; the literature on this topic is also limited [CL09,Cia10,PT07,CH17,BT17].…”

Optimal filtering of stochastic parabolic equations

“…The commutativity assumption ensures that the SPDE (1.1) is diagonalizable, that is, can be reduced to a system of uncoupled ordinary differential equations. Further work in that direction included analysis of maximum likelihood-type estimators for several scalar coefficients [11], sieve and kernel estimators for time-dependent coefficients θ = θ(t) [14,15], and Bayes-type estimators [3]. Non-diagonalizable models were also studied [16,30].…”