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错误发现率程序的运行特征与扩展AI 解读

Summary We investigate the operating characteristics of the Benjamini–Hochberg false discovery rate procedure for multiple testing. This is a distribution-free method that controls the expected fraction of falsely rejected null hypotheses among those rejected. The paper provides a framework for understanding more about this procedure. We first study the asymptotic properties of the ‘deciding point' D that determines the critical p-value. From this, we obtain explicit…

伯恩斯坦多项式后验的一致性AI 解读

Summary A Bernstein prior is a probability measure on the space of all the distribution functions on [0, 1]. Under very general assumptions, it selects absolutely continuous distribution functions, whose densities are mixtures of known beta densities. The Bernstein prior is of interest in Bayesian nonparametric inference with continuous data. We study the consistency of the posterior from a Bernstein prior. We first show that, under mild assumptions, the posterior is weakly…

后验分布的收敛速率AI 解读

We compute the rate at which the posterior distribution concentrates around the true parameter value. The spaces we work in are quite general and include in finite dimensional cases. The rates are driven by two quantities: the size of the space, as measured by bracketing entropy, and the degree to which the prior concentrates in a small ball around the true parameter. We consider two examples.

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高斯混合筛估计的收敛速度AI 解读

Gaussian mixtures provide a convenient method of density estimation that lies somewhere between parametric models and kernel density estimators.When the number of components of the mixture is allowed to increase as sample size increases, the model is called a mixture sieve.We establish a bound on the rate of convergence in Hellinger distance for density estimation using the Gaussian mixture sieve assuming that the true density is itself a mixture of Gaussians; the underlying…

使用数据依赖先验进行混合模型渐近推断AI 解读

Summary For certain mixture models, improper priors are undesirable because they yield improper posteriors. However, proper priors may be undesirable because they require subjective input. We propose the use of specially chosen data-dependent priors. We show that, in some cases, data-dependent priors are the only priors that produce intervals with second-order correct frequentist coverage. The resulting posterior also has another interpretation: it is the product of a fixed…

非参数问题中后验分布的一致性AI 解读

We give conditions that guarantee that the posterior probability of every Hellinger neighborhood of the true distribution tends to 1 almost surely. The conditions are (1) a requirement that the prior not put high mass near distributions with very rough densities and (2) a requirement that the prior put positive mass in Kullback-Leibler neighborhoods of the true distribution. The results are based on the idea of approximating the set of distributions with a finite-dimensional…

逮捕率的层级贝叶斯分析AI 解读

A Bayesian hierarchical model provides the basis for calibrating the crimes avoided by incarceration of individuals convicted of drug offenses compared to those convicted of nondrug offenses. Two methods for constructing reference priors for hierarchical models both lead to the same prior in the final model. We use Markov chain Monte Carlo methods to fit the model to data from a random sample of past arrest records of all felons convicted of drug trafficking, drug…

逮捕率的层级贝叶斯分析AI 解读

A Bayesian hierarchical model provides the basis for calibrating the crimes avoided by incarceration of individuals convicted of drug offenses compared to those convicted of nondrug offenses. Two methods for constructing reference priors for hierarchical models both lead to the same prior in the final model. We use Markov chain Monte Carlo methods to fit the model to data from a random sample of past arrest records of all felons convicted of drug trafficking, drug…

利用无限维指数族的贝叶斯拟合优度检验AI 解读

We develop a nonparametric Bayes factor for testing the fit of a parametric model. We begin with a nominal parametric family which we then embed into an infinite-dimensional exponential family. The new model then has a parametric and nonparametric component. We give the log density of the nonparametric component a Gaussian process prior. An asymptotic consistency requirement puts a restriction on the form of the prior, leaving us with a single hyperparameter for which we…

利用正态混合模型进行实用贝叶斯密度估计AI 解读

Mixtures of normals provide a flexible model for estimating densities in a Bayesian framework. There are some difficulties with this model, however. First, standard reference priors yield improper posteriors. Second, the posterior for the number of components in the mixture is not well defined (if the reference prior is used). Third, posterior simulation does not provide a direct estimate of the posterior for the number of components. We present some practical methods for…

结合模拟与渐近近似计算贝叶斯因子AI 解读

The Bayes factor is a ratio of two posterior normalizing constants, which may be difficult to compute. We compare several methods of estimating Bayes factors when it is possible to simulate observations from the posterior distributions, via Markov chain Monte Carlo or other techniques. The methods that we study are all easily applied without consideration of special features of the problem, provided that each posterior distribution is well behaved in the sense of having a…

贝叶斯推断的局部敏感性诊断AI 解读

We investigate diagnostics for quantifying the effect of small changes to the prior distribution over a k-dimensional parameter space. We show that several previously suggested diagnostics, such as the norm of the Fréchet derivative, diverge at rate $n^{k/2}$ if the base prior is an interior point in the class of priors, under the density ratio topology. Diagnostics based on $\phi$-divergences exhibit similar asymptotic behavior. We show that better asymptotic behavior can…

嵌套假设的参考贝叶斯检验及其与 Schwarz 准则的关系AI 解读

To compute a Bayes factor for testing H 0: ψ = ψ0 in the presence of a nuisance parameter β, priors under the null and alternative hypotheses must be chosen. As in Bayesian estimation, an important problem has been to define automatic, or “reference,” methods for determining priors based only on the structure of the model. In this article we apply the heuristic device of taking the amount of information in the prior on ψ equal to the amount of information in a single…

利用广义 Savage–Dickey 密度比计算贝叶斯因子AI 解读

We present a simple method for computing Bayes factors. The method derives from observing that in general, a Bayes factor can be written as the product of a quantity called the Savage-Dickey density ratio and a correction factor; both terms are easily estimated from posterior simulation. In some cases it is possible to do these computations without ever evaluating the likelihood.

无信息先验与干扰参数AI 解读

We study the conflict between priors that are noninformative for a parameter of interest versus priors that are noninformative for the whole parameter. Our investigation leads us to maximize a functional that has two terms: an asymptotic approximation to a standardized expected Kullback-Leibler distance between the marginal prior and marginal posterior for a parameter of interest, and a penalty term measuring the distance of the prior from the Jeffreys prior. A positive…

对称上概率AI 解读

As a first step toward developing statistical models based on upper and lower probabilities, we study upper probabilities and upper expectations on the unit interval that are symmetric, by which we mean invariant with respect to equimeasurability. These upper probabilities are generalizations of uniform probability measures. We give some characterizations of these upper probabilities. Specifically, we show that symmetry of the upper expectation functional is equivalent to…

期望界限的计算AI 解读

One method for evaluating the sensitivity of a Bayesian analysis is to embed the prior into a class of priors. Then bounds on prior and posterior quantities of interest must be computed. This approach to inference, often called robust Bayesian inference, has received much attention lately. Implementing robust Bayesian methods entails difficult computations, especially if the parameter space is high dimensional. In this article we develop a Monte Carlo approach to computing…

给定先验边际分布时的贝叶斯推断AI 解读

We show how to find bounds on posterior expectations of arbitrary functions of the parameters when the prior marginals are specified but when the complete joint prior is unspecified. We also give a theorem that is useful for finding posterior bounds in a wide range of Bayesian robustness problems. We apply these techniques to two examples. The first example involves a recent clinical trial for extracorporeal membrane oxygenation (ECMO). Our analysis may be regarded as a…

二元二项模型的先验分布AI 解读

We derive prior distributions for the bivariate binomial model using Bernardo's (1979) method. These priors are compared to the Jeifreys prior and to a prior proposed by Crowder & Sweeting (1989). The priors possess desirable symmetry properties since we allow them to depend on the parameter of interest.

二元二项模型的先验分布AI 解读

We derive prior distributions for the bivariate binomial model using Bernardo's (1979) method. These priors are compared to the Jeifreys prior and to a prior proposed by Crowder & Sweeting (1989). The priors possess desirable symmetry properties since we allow them to depend on the parameter of interest.