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采用新先验族的多元正态协方差矩阵贝叶斯分析AI 解读

Bayesian analysis for the covariance matrix of a multivariate normal distribution has received a lot of attention in the last two decades. In this paper, we propose a new class of priors for the covariance matrix, including both inverse Wishart and reference priors as special cases. The main motivation for the new class is to have available priors—both subjective and objective—that do not “force eigenvalues apart,” which is a criticism of inverse Wishart and Jeffreys priors.…

稳健的高斯随机过程仿真代理建模AI 解读

We consider estimation of the parameters of a Gaussian Stochastic Process (GaSP), in the context of emulation (approximation) of computer models for which the outcomes are real-valued scalars. The main focus is on estimation of the GaSP parameters through various generalized maximum likelihood methods, mostly involving finding posterior modes; this is because full Bayesian analysis in computer model emulation is typically prohibitively expensive. The posterior modes that are…

变量选择中贝叶斯与经验贝叶斯的多重性调整AI 解读

This paper studies the multiplicity-correction effect of standard Bayesian variable-selection priors in linear regression. Our first goal is to clarify when, and how, multiplicity correction happens automatically in Bayesian analysis, and to distinguish this correction from the Bayesian Ockham’s-razor effect. Our second goal is to contrast empirical-Bayes and fully Bayesian approaches to variable selection through examples, theoretical results and simulations. Considerable…

正态层级模型中后验的适当性与超先验的可容许性AI 解读

Hierarchical modeling is wonderful and here to stay, but hyperparameter priors are often chosen in a casual fashion. Unfortunately, as the number of hyperparameters grows, the effects of casual choices can multiply, leading to considerably inferior performance. As an extreme, but not uncommon, example use of the wrong hyperparameter priors can even lead to impropriety of the posterior. For exchangeable hierarchical multivariate normal models, we first determine when a…

客观贝叶斯模型选择中的训练样本AI 解读

Central to several objective approaches to Bayesian model selection is the use of training samples (subsets of the data), so as to allow utilization of improper objective priors. The most common prescription for choosing training samples is to choose them to be as small as possible, subject to yielding proper posteriors; these are called minimal training samples. When data can vary widely in terms of either information content or impact on the improper priors, use of minimal…

最优预测模型选择AI 解读

Often the goal of model selection is to choose a model for future prediction, and it is natural to measure the accuracy of a future prediction by squared error loss. Under the Bayesian approach, it is commonly perceived that the optimal predictive model is the model with highest posterior probability, but this is not necessarily the case. In this paper we show that, for selection among normal linear models, the optimal predictive model is often the median probability model,…

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复合原假设模型的p值AI 解读

The problem of investigating compatibility of an assumed model with the data is investigated in the situation when the assumed model has unknown parameters. The most frequently used measures of compatibility are p values, based on statistics T for which large values are deemed to indicate incompatibility of the data and the model. When the null model has unknown parameters, p values are not uniquely defined. The proposals for computing a p value in such a situation include…

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层次先验的选择：正态均值估计的可容许性AI 解读

In hierarchical Bayesian modeling of normal means, it is common to complete the prior specification by choosing a constant prior density for unmodeled hyperparameters (e.g., variances and highest-level means). This common practice often results in an inadequate overall prior, inadequate in the sense that estimators resulting from its use can be inadmissible under quadratic loss. In this paper, hierarchical priors for normal means are categorized in terms of admissibility and…

用于模型选择与预测的内在贝叶斯因子AI 解读

In the Bayesian approach to model selection or hypothesis testing with models or hypotheses of differing dimensions, it is typically not possible to utilize standard noninformative (or default) prior distributions. This has led Bayesians to use conventional proper prior distributions or crude approximations to Bayes factors. In this article we introduce a new criterion called the intrinsic Bayes factor, which is fully automatic in the sense of requiring only standard…

使用参考先验估计协方差矩阵AI 解读

Estimation of a covariance matrix $\sum$ is a notoriously difficult problem; the standard unbiased estimator can be substantially suboptimal. We approach the problem from a noninformative prior Bayesian perspective, developing the reference noninformative prior for a covariance matrix and obtaining expressions for the resulting Bayes estimators. These expressions involve the computation of high-dimensional posterior expectations, which is done using a recent Markov chain…

AR(1) 模型的无信息先验与贝叶斯检验AI 解读

Various approaches to the development of a noninformative prior for the AR(1) model are considered and compared. Particular attention is given to the reference prior approach, which seems to work well for the stationary case but encounters difficulties in the explosive case. A symmetrized (proper) version of the stationary reference prior is ultimately recommended for the problem. Bayesian testing of the unit root, stationary, and explosive hypotheses is considered also.…

Poly-Weibull 分布的贝叶斯分析AI 解读

In this article Bayesian analysis for a Poly-Weibull distribution using informative priors is discussed. This distribution typically arises when the data is the minimum of several Weibull failure times from competing risks. To perform the Bayesian computations, simulation using the Gibbs sampler is suggested. This can be used to find posterior moments, the marginal posterior probability density function, and the predictive risk or reliability.

指数回归模型推断的无信息先验AI 解读

In the exponential regression model, inference concerning the regression parameter is notoriously difficult, even when using the Bayesian noninformative prior approach. The reference prior approach (Bernardo, 1979; Berger & Bernardo, 1989) is considered, and argued to yield very satisfactory inferences. Estimation and credible sets are considered in a specific example.

In the exponential regression model, inference concerning the regression parameter is notoriously difficult, even when using the Bayesian noninformative prior approach. The reference prior approach (Bernardo, 1979; Berger & Bernardo, 1989) is considered, and argued to yield very satisfactory inferences. Estimation and credible sets are considered in a specific example.

多项分布贝叶斯因子的下界及其在卡方拟合检验中的应用AI 解读

Lower bounds on Bayes factors in favor of the null hypothesis in multinomial tests of point null hypotheses are developed. These are then applied to derive lower bounds on Bayes factors in both exact and asymptotic chi-squared testing situations. The general conclusion is that the lower bounds tend to be substantially larger than $P$-values, raising serious questions concerning the routine use of moderately small $P$-values (e.g., 0.05) to represent significant evidence…

多元正态均值的主观层次贝叶斯估计：与频率学派的联系AI 解读

In shrinkage estimation of a multivariate normal mean, the two dominant approaches to construction of estimators have been the hierarchical or empirical Bayes approach and the minimax approach. The first has been most extensively used in practice, because of its greater flexibility in adapting to varying situations, while the second has seen the most extensive theoretical development. In this paper we consider several topics on the interface of these approaches,…

正态均值估计：损失的频率学派估计AI 解读

In estimation of a $p$-variate normal mean with identity covariance matrix, Stein-type estimators can offer significant gains over the $\operatorname{mle}$ in terms of risk with respect to sum of squares error loss. Their maximum risk is still equal to $p$, however, which will typically be their "reported loss." In this paper we consider use of data-dependent "loss estimators." Two conditions that are attractive for such a loss estimator are that it be an improved loss…

单峰污染先验下后验量的取值范围AI 解读

We consider the problem of robustness or sensitivity of given Bayesian posterior criteria to specification of the prior distribution. Criteria considered include the posterior mean, variance and probability of a set (for credible regions and hypothesis testing). Uncertainty in an elicited prior, $\pi_0$, is modelled by an $\varepsilon$-contamination class $\Gamma = \{\pi = (1 - \varepsilon)\pi_0 + \varepsilon q, q \in Q\}$, where $\varepsilon$ reflects the amount of…

均值乘积的估计：基于参考先验的贝叶斯分析AI 解读

Suppose that we observe X ∼ N(α, 1) and, independently, Y ∼ N(β, 1), and are concerned with inference (mainly estimation and confidence statements) about the product of means θ = αβ. This problem arises, most obviously, in situations of determining area based on measurements of length and width. It also arises in other practical contexts, however. For instance, in gypsy moth studies, the hatching rate of larvae per unit area can be estimated as the product of the mean of egg…

指定分位数的拟单峰先验下后验概率的范围AI 解读

Suppose several quantiles of the prior distribution for θ are specified or, equivalently, the prior probabilities of a partitioning collection of intervals {Ii } are given. In addition, suppose that the prior distribution is assumed to be unimodal. Rather than selecting a single prior distribution to perform a Bayesian analysis, it is of interest to consider the class of all prior distributions compatible with these inputs. For this class and unimodal likelihood functions,…

给定分位数的拟单峰先验下后验概率的取值范围AI 解读

Suppose several quantiles of the prior distribution for θ are specified or, equivalently, the prior probabilities of a partitioning collection of intervals {Ii } are given. In addition, suppose that the prior distribution is assumed to be unimodal. Rather than selecting a single prior distribution to perform a Bayesian analysis, it is of interest to consider the class of all prior distributions compatible with these inputs. For this class and unimodal likelihood functions,…

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