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二维 Navier–Stokes 方程的数据同化：后验测度的最优高斯渐近性质AI 解读

A functional Bernstein–von Mises theorem is proved for posterior measures arising in a data assimilation problem with the two-dimensional Navier–Stokes equation where a Gaussian process prior is assigned to the initial condition of the system. The posterior measure, which provides the update in the space of all trajectories arising from a discrete sample of the (deterministic) dynamics, is shown to be approximated by a Gaussian random vector field obtained as the solution to…

从杀死扩散过程推断扩散率AI 解读

We consider diffusion of independent molecules in an insulated Euclidean domain with unknown diffusivity parameter. At a random time and position, the molecules may bind and stop diffusing in dependence of a given “binding potential.” The binding process can be modeled by an additive random functional corresponding to the canonical construction of a “killed” diffusion Markov process. We study the problem of conducting inference on the infinite-dimensional diffusion parameter…

McKean–Vlasov 模型的贝叶斯非参数推断AI 解读

We consider nonparametric statistical inference on a periodic interaction potential W from noisy discrete space-time measurements of solutions ρ=ρW of the nonlinear McKean–Vlasov equation, describing the probability density of the mean field limit of an interacting particle system. We show how Gaussian process priors assigned to W give rise to posterior mean estimators that exhibit fast convergence rates for the implied estimated densities ρ¯ towards ρW. We further show that…

高斯过程先验下数据同化的后验一致性：二维Navier–Stokes方程AI 解读

We consider a nonlinear Bayesian data assimilation model for the periodic two-dimensional Navier–Stokes equations with initial condition modelled by a Gaussian process prior. We show that if the system is updated with sufficiently many discrete noisy measurements of the velocity field, then the posterior distribution eventually concentrates near the ground truth solution of the time evolution equation, and in particular that the initial condition is recovered consistently by…

基于低频观测的扩散过程一致推断AI 解读

Let (Xt) be a reflected diffusion process in a bounded convex domain in Rd, solving the stochastic differential equation dXt=∇f(Xt)dt+ 2f(Xt)dWt,t≥0, with Wt a d-dimensional Brownian motion. The data X0,XD,…,XND consist of discrete measurements and the time interval D between consecutive observations is fixed so that one cannot ‘zoom’ into the observed path of the process. The goal is to infer the diffusivity f and the associated transition operator Pt,f. We prove…

多维扩散过程漂移向量场的非参数统计推断AI 解读

The problem of determining a periodic Lipschitz vector field $b=(b_{1},\ldots ,b_{d})$ from an observed trajectory of the solution $(X_{t}:0\le t\le T)$ of the multi-dimensional stochastic differential equation \begin{equation*}dX_{t}=b(X_{t})\,dt+dW_{t},\quad t\geq 0,\end{equation*} where $W_{t}$ is a standard $d$-dimensional Brownian motion, is considered. Convergence rates of a penalised least squares estimator, which equals the maximum a posteriori (MAP) estimate…

主成分线性泛函的有效估计AI 解读

We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations $X_{1},\dots,X_{n}$ in a separable Hilbert space $\mathbb{H}$ with unknown covariance operator $\Sigma $. The complexity of the problem is characterized by its effective rank $\mathbf{r}(\Sigma):=\frac{\operatorname{tr}(\Sigma)}{\|\Sigma \|}$, where $\mathrm{tr}(\Sigma)$ denotes the trace of $\Sigma $ and $\|\Sigma\|$ denotes its operator norm. We develop a method of bias reduction in the…

We consider the statistical inverse problem of recovering a function $f:M\to \mathbb{R}$, where $M$ is a smooth compact Riemannian manifold with boundary, from measurements of general $X$-ray transforms $I_{a}(f)$ of $f$, corrupted by additive Gaussian noise. For $M$ equal to the unit disk with “flat” geometry and $a=0$ this reduces to the standard Radon transform, but our general setting allows for anisotropic media $M$ and can further model local “attenuation” effects—both…

离散观测标量扩散的非参数贝叶斯后验收缩率AI 解读

We consider nonparametric Bayesian inference in a reflected diffusion model $dX_{t}=b(X_{t})\,dt+\sigma(X_{t})\,dW_{t}$, with discretely sampled observations $X_{0},X_{\Delta},\ldots,X_{n\Delta}$. We analyse the nonlinear inverse problem corresponding to the “low frequency sampling” regime where $\Delta>0$ is fixed and $n\to\infty$. A general theorem is proved that gives conditions for prior distributions $\Pi$ on the diffusion coefficient $\sigma$ and the drift function $b$…

摘要暂缺

非参数贝叶斯程序中的 Bernstein–von Mises 现象AI 解读

We continue the investigation of Bernstein–von Mises theorems for nonparametric Bayes procedures from [Ann. Statist. 41 (2013) 1999–2028]. We introduce multiscale spaces on which nonparametric priors and posteriors are naturally defined, and prove Bernstein–von Mises theorems for a variety of priors in the setting of Gaussian nonparametric regression and in the i.i.d. sampling model. From these results we deduce several applications where posterior-based inference coincides…

稀疏回归中的置信集AI 解读

The problem of constructing confidence sets in the high-dimensional linear model with $n$ response variables and $p$ parameters, possibly $p\ge n$, is considered. Full honest adaptive inference is possible if the rate of sparse estimation does not exceed $n^{-1/4}$, otherwise sparse adaptive confidence sets exist only over strict subsets of the parameter spaces for which sparse estimators exist. Necessary and sufficient conditions for the existence of confidence sets that…

高斯白噪声模型中的非参数 Bernstein–von Mises 定理AI 解读

Bernstein–von Mises theorems for nonparametric Bayes priors in the Gaussian white noise model are proved. It is demonstrated how such results justify Bayes methods as efficient frequentist inference procedures in a variety of concrete nonparametric problems. Particularly Bayesian credible sets are constructed that have asymptotically exact $1-\alpha$ frequentist coverage level and whose $L^{2}$-diameter shrinks at the minimax rate of convergence (within logarithmic factors)…

L^r 距离下后验分布的收缩速率（1≤r≤∞）AI 解读

The frequentist behavior of nonparametric Bayes estimates, more specifically, rates of contraction of the posterior distributions to shrinking Lr-norm neighborhoods, 1 ≤ r ≤ ∞, of the unknown parameter, are studied. A theorem for nonparametric density estimation is proved under general approximation-theoretic assumptions on the prior. The result is applied to a variety of common examples, including Gaussian process, wavelet series, normal mixture and histogram priors. The…

自适应推断与置信带AI 解读

The problem of existence of adaptive confidence bands for an unknown density f that belongs to a nested scale of Hölder classes over ℝ or [0, 1] is considered. Whereas honest adaptive inference in this problem is impossible already for a pair of Hölder balls Σ(r), Σ(s), r ≠ s, of fixed radius, a nonparametric distinguishability condition is introduced under which adaptive confidence bands can be shown to exist. It is further shown that this condition is necessary and…

小波反卷积估计量的全局一致风险界AI 解读

We consider the statistical deconvolution problem where one observes n replications from the model Y = X + ϵ, where X is the unobserved random signal of interest and ϵ is an independent random error with distribution φ. Under weak assumptions on the decay of the Fourier transform of φ, we derive upper bounds for the finite-sample sup-norm risk of wavelet deconvolution density estimators fn for the density f of X, where f : ℝ → ℝ is assumed to be bounded. We then derive lower…

密度估计中的置信带AI 解读

Given a sample from some unknown continuous density f : ℝ→ℝ, we construct adaptive confidence bands that are honest for all densities in a “generic” subset of the union of t-Hölder balls, 0