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非参数判别中趋近贝叶斯风险的最优收敛速率AI 解读

Consider the multiclassification (discrimination) problem with known prior probabilities and a multi-dimensional vector of observations. Assume the underlying densities corresponding to the various classes are unknown but a training sample of size $N$ is available from each class. Rates of convergence to Bayes risk are investigated under smoothness conditions on the underlying densities of the type often seen in nonparametric density estimation. These rates can be…

贝叶斯概率评估者的校准、充分性与支配关系AI 解读

DeGroot and Fienberg (1982a) recently considered various aspects of the problem of evaluating the performance of probability appraisers. After briefly reviewing their notions of calibration and sufficiency we introduce related ideas of semicalibration and domination and consider their relationship to the earlier concepts. We then discuss some simple Bayesian mechanisms for making probability assessments and study their calibration, semicalibration, sufficiency, and…

纵向研究中变化率的经验贝叶斯估计AI 解读

The usually irregular follow-up intervals in epidemiologic studies preclude the use of classical growth curve analysis. For short follow-up intervals, it is suggested that individual rates of change are useful for exploratory analysis. Empirical Bayes estimates of these rates of change are recommended and developed. An example of bone loss with age in women is also given.

贝叶斯概率评估的校准、充分性与支配关系AI 解读

DeGroot and Fienberg (1982a) recently considered various aspects of the problem of evaluating the performance of probability appraisers. After briefly reviewing their notions of calibration and sufficiency we introduce related ideas of semicalibration and domination and consider their relationship to the earlier concepts. We then discuss some simple Bayesian mechanisms for making probability assessments and study their calibration, semicalibration, sufficiency, and…

交叉验证平滑样条的贝叶斯“置信区间”AI 解读

SUMMARY We consider the model Y(ti)=g(ti) + ∈i, i = 1, 2, …, n, where g(t), t ∈ [0, 1] is a smooth function and the {∈i} are independent N(0, σ2) errors with σ2 unknown. The cross-validated smoothing spline can be used to estimate g non-parametrically from observations on Y(ti), i= 1, 2, …, n, and the purpose of this paper is to study confidence intervals for this estimate. Properties of smoothing splines as Bayes estimates are used to derive confidence intervals based on…

控制问题中不可容许估计量的改进AI 解读

Let $X$ have a $p$-variate normal distribution with unknown mean $\theta$ and identity covariance matrix. The following transformed version of a control problem (Zaman, 1981) is considered: estimate $\theta$ by $d$ subject to incurring a loss $L(d, \theta) = (\theta^t d - 1)^2$. The comparison of decision rules in terms of expected loss is reduced to the study of differential inequalities. Results establishing the minimaxity of a large class of estimators are obtained.…

关于经验贝叶斯估计的若干思考AI 解读

Examples are given to illustrate the "linear" and "general" empirical Bayes approaches to estimation. A final example concerns testing the null hypothesis that a treatment has had no effect.

利用狄利克雷混合分布估计二乘二列联表的贝叶斯方法AI 解读

In the estimation of cell probabilities from a 2 × 2 table, a prior distribution is developed that can reflect prior beliefs about the cross-classification structure in the table. The posterior means using this prior shrink the classical estimates towards the association structure specified a priori by the user. Closed-form approximations to the posterior means and credible regions are developed in the special case where the two variables of the table are believed…

多元正态分布均值的经验贝叶斯置信集AI 解读

Through the use of an empirical Bayes argument, a confidence set for the mean of a multivariate normal distribution is derived. The set is a recentered sphere, is easy to compute, and has uniformly smaller volume than the usual confidence set. An exact formula for the coverage probability is derived, and numerical evidence is presented which shows that the empirical Bayes set uniformly dominates the usual set in coverage probability.

多重超几何函数的概率解释与统计用途AI 解读

This article reviews and interprets recent mathematics of special functions, with emphasis on integral representations of multiple hypergeometric functions. B.C. Carlson's centrally important parameterized functions R and ℛ, initially defined as Dirichlet averages, are expressed as probability-generating functions of mixed multinomial distributions. Various nested families generalizing the Dirichlet distributions are developed for Bayesian inference in multinomial sampling…

线性贝叶斯估计量的一般方差修正AI 解读

We consider the problem of modifying the linear Bayes estimator for the mean of a distribution of unknown form by using an estimate of sample variance. The general conditions under which there is no advantageous variance modification to the linear Bayes rule are identified. These results are applied to the problem of sampling from a normal distribution with unknown mean and variance, and the implications are discussed.

无偏性作为贝叶斯性的对偶AI 解读

In this note we define the notion of unbiasedness for a decision function for an arbitrary loss function. This is a generalization of Lehmann's (1951) definition. We show that this notion of unbiasedness is a dual to the notion of being Bayes; that is, if the role of the random variable and the parameter is interchanged, then unbiasedness is equivalent to being Bayes and vice versa. Some consequences of this fact are discussed.

摘要暂缺

线性贝叶斯估计量的一般方差修正AI 解读

We consider the problem of modifying the linear Bayes estimator for the mean of a distribution of unknown form by using an estimate of sample variance. The general conditions under which there is no advantageous variance modification to the linear Bayes rule are identified. These results are applied to the problem of sampling from a normal distribution with unknown mean and variance, and the implications are discussed.

摘要暂缺

贝叶斯非线性回归及一个酶动力学实例AI 解读

A nonlinear regression model is applied to several sets of enzyme kinetics data, treating the entire regression vector as the parameter of interest. The resulting marginal posterior distributions are presented alongside the usual Student's t posterior distributions implicit in the usual linearized regression. The prior distribution used is that which is minimally informative about the regression vector. In the special case of the linear model this procedure leads to the…

响应变量存在缺失观测的多元回归贝叶斯推断AI 解读

We discuss the case of the multivariate linear model Y = XB + E with Y an (n × p) matrix, and so on, when there are missing observations in the Y matrix in a so-called nested pattern. We propose an analysis that arises by incorporating the predictive density of the missing observations in determining the posterior distribution of B, and its mean and variance matrix. This involves us with matric-T variables. The resulting analysis is illustrated with some Canadian economic…

Ferguson 分布函数贝叶斯估计量的稳健性AI 解读

We derive an explicit expression for the Bayes risk (using weighted squared error loss) of Dalal's Bayes estimator of a symmetric distribution under a $\mathscr{G}$-invariant Dirichlet process prior. We compare this risk to the risk of Ferguson's estimator of an arbitrary distribution under the $\mathscr{G}$-invariant prior. This enables us to (i) assess the savings in risk attained by incorporating known symmetry structure in the model and (ii) provide information about the…

The normal, Poisson, gamma, binomial, negative binomial, and NEFGHS distributions are the six univariate natural exponential families (NEF) with quadratic variance functions (QVF). This sequel to Morris (1982) treats certain statistical topics that can be handled within this unified NEF-QVF formulation, including unbiased estimation, Bhattacharyya and Cramer-Rao lower bounds, conditional distributions and moments, quadratic regression, conjugate prior distributions, moments…

正确选择概率的 Olkin-Sobel-Tong 估计量的小样本性质AI 解读

In the problem of selecting the best of k populations, Olkin, Sobel, and Tong (1976) have introduced the important idea of a posteriori analysis of the data (as opposed to the usual formulation), in which design of the experiment is the major consideration. They considered the large-sample properties of an estimator that has been discussed further by Gibbons, Olkin, and Sobel (1977), Gupta and Panchapakesan (1979), and Tong (1980). In this article we study the small-sample…

作为证据的先验与似然比AI 解读

Arguments based on diagnostic data concerning a particular case, and ones based on prior experience with like cases, can be represented by belief functions and combined by Dempster's Rule. In limiting cases, the belief functions depend on likelihood ratio or on prior odds, and when these limiting cases occur together, Bayes' Theorem is applicable as is Dempster's Rule. The resulting functional equations lead to a single-parameter family of functions relating belief strength…

贝叶斯同时估计中的序贯分配与可选停止AI 解读

Suppose random variables X 1, X 2, …, with distribution depending on parameter θ1, are observable from population 1, independent of random variables Y 1, Y 2, …, with distribution depending on parameter θ2, and observed from population 2. The simultaneous estimation of parameters θ1 and θ2 is of interest. An experiment is conducted where at each stage either an X or a Y is observed, and the experiment may be stopped at any stage (allocation and decision depending on…

结合人类与其他物种癌症研究结果的贝叶斯方法AI 解读

We propose a class of Bayesian statistical methods for interspecies extrapolation of dose-response functions. The methods distinguish formally between the conventional sampling error within each dose-response experiment and a novel error of uncertain relevance between experiments. Through a system of hierarchical prior distributions similar to that of Lindley and Smith (1972), the dose-response data from many substances and species are used to estimate the interexperimental…

信息度量与贝叶斯层级模型AI 解读

We consider the Bayesian statistical models in which the prior distribution of the parameter vector θ1 in the distribution of an observable random vector X is to be specified in a hierarchical fashion and one wants to learn about the hyperparameters at each level of this prior distribution. It is shown that for a wide class of information measures, based on the so-called f divergence, the information decreases as one moves to higher levels of hyperparameters. This result…