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Wicksell 问题中经保序化后验的半参数 Bernstein–von Mises 现象AI 解读

In this paper, we propose a novel Bayesian approach for nonparametric estimation in Wicksell’s problem. This has important applications in astronomy for estimating the distribution of the positions of the stars in a galaxy given projected stellar positions and in materials science to determine the 3D microstructure of a material, using its 2D cross-sections. We deviate from the classical Bayesian nonparametric approach, which would place a Dirichlet Process (DP) prior on the…

利用空间分布式高斯过程实现自适应AI 解读

We consider the accuracy of an approximate posterior distribution in nonparametric regression problems by combining posterior distributions computed on subsets of the data defined by the locations of the independent variables. We show that this approximate posterior retains the rate of recovery of the full data posterior distribution, where the rate of recovery adapts to the smoothness of the true regression function. As particular examples we consider Gaussian process…

半参数贝叶斯因果推断AI 解读

We develop a semiparametric Bayesian approach for estimating the mean response in a missing data model with binary outcomes and a nonparametrically modelled propensity score. Equivalently, we estimate the causal effect of a treatment, correcting nonparametrically for confounding. We show that standard Gaussian process priors satisfy a semiparametric Bernstein–von Mises theorem under smoothness conditions. We further propose a novel propensity score-dependent prior that…

结构化高维模型中泛函的极小极大估计AI 解读

We introduce a new method of estimation of parameters in semiparametric and nonparametric models. The method employs $U$-statistics that are based on higher-order influence functions of the parameter of interest, which extend ordinary linear influence functions, and represent higher derivatives of this parameter. For parameters for which the representation cannot be perfect the method often leads to a bias-variance trade-off, and results in estimators that converge at a…

采用稀疏先验的贝叶斯线性回归AI 解读

We study full Bayesian procedures for high-dimensional linear regression under sparsity constraints. The prior is a mixture of point masses at zero and continuous distributions. Under compatibility conditions on the design matrix, the posterior distribution is shown to contract at the optimal rate for recovery of the unknown sparse vector, and to give optimal prediction of the response vector. It is also shown to select the correct sparse model, or at least the coefficients…

大海捞针：可能稀疏序列的后验集中AI 解读

We consider full Bayesian inference in the multivariate normal mean model in the situation that the mean vector is sparse. The prior distribution on the vector of means is constructed hierarchically by first choosing a collection of nonzero means and next a prior on the nonzero values. We consider the posterior distribution in the frequentist set-up that the observations are generated according to a fixed mean vector, and are interested in the posterior distribution of the…

光滑密度下 Dirichlet 混合模型的后验收敛率AI 解读

We study the rates of convergence of the posterior distribution for Bayesian density estimation with Dirichlet mixtures of normal distributions as the prior. The true density is assumed to be twice continuously differentiable. The bandwidth is given a sequence of priors which is obtained by scaling a single prior by an appropriate order. In order to handle this problem, we derive a new general rate theorem by considering a countable covering of the parameter space whose…

非独立同分布观测下后验分布的收敛速度AI 解读

We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion. We then specialize our results to independent, nonidentically distributed observations, Markov processes, stationary Gaussian time series and the white noise model. We…

自适应非参数置信集AI 解读

We construct honest confidence regions for a Hilbert space-valued parameter in various statistical models. The confidence sets can be centered at arbitrary adaptive estimators, and have diameter which adapts optimally to a given selection of models. The latter adaptation is necessarily limited in scope. We review the notion of adaptive confidence regions, and relate the optimal rates of the diameter of adaptive confidence regions to the minimax rates for testing and…

复合原假设模型中 p 值的渐近分布AI 解读

We investigate the compatibility of a null model H 0 with the data by calculating a p value; that is, the probability, under H 0, that a given test statistic T exceeds its observed value. When the null model consists of a single distribution, the p value is readily obtained, and it has a uniform distribution under H 0. On the other hand, when the null model depends on an unknown nuisance parameter θ one must somehow get rid of θ, (e.g., by estimating it) to calculate a p…

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空间抽样方案下的最大似然估计AI 解读

It is shown that the maximum likelihood estimator in a model used in the statistical analysis of computer experiments is asymptotically efficient.

半参数混合模型中的有效最大似然估计AI 解读

We consider maximum likelihood estimation in several examples of semiparametric mixture models, including the exponential frailty model and the errors-in-variables model. The observations consist of a sample of size n from the mixture density $\int p_{\theta}(x|z) d \eta(z)$. The mixing distribution is completely unknown. We show that the first component $\hat{\theta}_n$ of the joint maximum likelihood estimator , $(\hat{\theta}_n \hat{\eta}_n)$ is asymptotically normal and…

部分删失数据下的最大似然估计AI 解读

Suppose one observes independent samples of size $n$ from both the mixture density $\int p(x \mid z) d\eta(z)$ and from the distribution $\eta$. The kernel $p(x \mid z)$ is known. We show asymptotic normality and efficiency of the maximum likelihood estimator for $\eta$.

渐近信息界AI 解读

This paper discusses several lower bound results for the asymptotic performance of estimators of smooth functionals in i.i.d. models. The key idea is to look at a set of local limiting distributions of an estimator sequence, rather than to impose regularity conditions, or to consider limits of maximum risk. Special attention is paid to situations where the tangent cone is not a linear space. As an example, the local asymptotic minimax risk in mixture models is computed.