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We revisit a paper by Charles Stein, and discuss its follow-up.

自适应非参数贝叶斯可信集的频率学覆盖率AI 解读

We investigate the frequentist coverage of Bayesian credible sets in a nonparametric setting. We consider a scale of priors of varying regularity and choose the regularity by an empirical Bayes method. Next we consider a central set of prescribed posterior probability in the posterior distribution of the chosen regularity. We show that such an adaptive Bayes credible set gives correct uncertainty quantification of “polished tail” parameters, in the sense of high probability…

采用高斯先验的贝叶斯逆问题AI 解读

The posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter, and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the minimax rate. The frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter, with smoother priors leading to zero coverage and rougher priors to conservative coverage.…

使用逆 Gamma 带宽高斯随机场的自适应贝叶斯估计AI 解读

We consider nonparametric Bayesian estimation inference using a rescaled smooth Gaussian field as a prior for a multidimensional function. The rescaling is achieved using a Gamma variable and the procedure can be viewed as choosing an inverse Gamma bandwidth. The procedure is studied from a frequentist perspective in three statistical settings involving replicated observations (density estimation, regression and classification). We prove that the resulting posterior…

基于高斯过程先验的后验分布收缩速率AI 解读

We derive rates of contraction of posterior distributions on nonparametric or semiparametric models based on Gaussian processes. The rate of contraction is shown to depend on the position of the true parameter relative to the reproducing kernel Hilbert space of the Gaussian process and the small ball probabilities of the Gaussian process. We determine these quantities for a range of examples of Gaussian priors and in several statistical settings. For instance, we consider…

无限维贝叶斯统计中的模型误设AI 解读

We consider the asymptotic behavior of posterior distributions if the model is misspecified. Given a prior distribution and a random sample from a distribution P0, which may not be in the support of the prior, we show that the posterior concentrates its mass near the points in the support of the prior that minimize the Kullback–Leibler divergence with respect to P0. An entropy condition and a prior-mass condition determine the rate of convergence. The method is applied to…

论剖面似然AI 解读

We show that semiparametric profile likelihoods, where the nuisance parameter has been profiled out, behave like ordinary likelihoods in that they have a quadratic expansion. In this expansion the score function and the Fisher information are replaced by the efficient score function and efficient Fisher information. The expansion may be used, among others, to prove the asymptotic normality of the maximum likelihood estimator, to derive the asymptotic chi-squared distribution…

比例优势模型的最大似然估计AI 解读

We consider maximum likelihood estimation of the parameters in the proportional odds model with right-censored data. The estimator of the regression coefficient is shown to be asymptotically normal with efficient variance. The maximum likelihood estimator of the unknown monotonic transformation of the survival time converges uniformly at a parametric rate to the true transformation. Estimates for the standard errors of the estimated regression coefficients are obtained by…

半参数似然比推断AI 解读

Likelihood ratio tests and related confidence intervals for a real parameter in the presence of an infinite dimensional nuisance parameter are considered. In all cases, the estimator of the real parameter has an asymptotic normal distribution. However, the estimator of the nuisance parameter may not be asymptotically Gaussian or may converge to the true parameter value at a slower rate than the square root of the sample size. Nevertheless the likelihood ratio statistic is…

一类半参数模型中实值参数的估计AI 解读

We study semiparametric models where for a fixed value of the finite-dimensional parameter there exists a sufficient statistic for the nuisance parameter. An asymptotically normal sequence of estimators for the parametric component is constructed, which is efficient under the assumption that projecting on the set of nuisance scores is equivalent to taking conditional expectations given the sufficient statistic. The latter property is checked for a number of examples, in…