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贝叶斯 Probit 回归中 Albert–Chib 算法的收敛复杂度分析AI 解读

The use of MCMC algorithms in high dimensional Bayesian problems has become routine. This has spurred so-called convergence complexity analysis, the goal of which is to ascertain how the convergence rate of a Monte Carlo Markov chain scales with sample size, $n$, and/or number of covariates, $p$. This article provides a thorough convergence complexity analysis of Albert and Chib’s [J. Amer. Statist. Assoc. 88 (1993) 669–679] data augmentation algorithm for the Bayesian…

采用非正常先验的贝叶斯一般线性混合模型 Gibbs 采样器收敛分析AI 解读

Bayesian analysis of data from the general linear mixed model is challenging because any nontrivial prior leads to an intractable posterior density. However, if a conditionally conjugate prior density is adopted, then there is a simple Gibbs sampler that can be employed to explore the posterior density. A popular default among the conditionally conjugate priors is an improper prior that takes a product form with a flat prior on the regression parameter, and so-called power…

通过马尔可夫链论证评价形式后验分布AI 解读

We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function φ of a parameter when the loss is quadratic. If the posterior mean of φ is admissible for all bounded φ, the posterior is strongly admissible. We give sufficient conditions for strong admissibility. These conditions involve the recurrence of a Markov chain associated with the estimation problem. We develop general sufficient…

利用目标函数与随机搜索进行聚类AI 解读

Summary A new approach to clustering multivariate data, based on a multilevel linear mixed model, is proposed. A key feature of the model is that observations from the same cluster are correlated, because they share cluster-specific random effects. The inclusion of cluster-specific random effects allows parsimonious departure from an assumed base model for cluster mean profiles. This departure is captured statistically via the posterior expectation, or best linear unbiased…

贝叶斯Probit回归MCMC算法的收敛速度与渐近标准误AI 解读

Summary Consider a probit regression problem in which Y1, …, Yn are independent Bernoulli random variables such that Pr(Yi=1)=Φ(xiTβ) where xi is a p-dimensional vector of known covariates that are associated with Yi, β is a p-dimensional vector of unknown regression coefficients and Φ(·) denotes the standard normal distribution function. We study Markov chain Monte Carlo algorithms for exploring the intractable posterior density that results when the probit regression…

层级随机效应模型中 Gibbs 采样的充分预热长度AI 解读

We consider Gibbs and block Gibbs samplers for a Bayesian hierarchical version of the one-way random effects model. Drift and minorization conditions are established for the underlying Markov chains. The drift and minorization are used in conjunction with results from J. S. Rosenthal [J. Amer. Statist. Assoc. 90 (1995) 558–566] and G. O. Roberts and R. L. Tweedie [Stochastic Process. Appl. 80 (1999) 211–229] to construct analytical upper bounds on the distance to…

非负整数马尔可夫链的常返与暂留条件及几何成功概率估计AI 解读

Let $Z$ be a discrete random variable with support $\Z^+ = \{0,1,2,\dots\}$. We consider a Markov chain $Y=(Y_n)_{n=0}^\infty$ with state space $\Z^+$ and transition probabilities given by $P(Y_{n+1} = j|Y_n = i) = P(Z = i+j)/P(Z \geq i)$. We prove that convergence of $\sum_{n=1}^\infty 1/[n^3 P (Z=n)]$ is sufficient for transience of $Y$ while divergence of $\sum_{n=1}^\infty 1/[n^2 P (Z \geq n)]$ is sufficient for recurrence. Let $X$ be a $\mbox{Geometric}(p)$ random…

层级模型：当时的计算视角AI 解读

Hierarchical models (HMs) provide a flexible framework for modeling data. The ongoing development of techniques like the EM algorithm and Markov chain Monte Carlo has enabled statisticians to make use of increasingly more complicated HMs over the last few decades. In this article, we consider Bayesian and frequentist versions of a general, two-stage HM, and describe several examples from the literature that illustrate its versatility. Some key aspects of the computational…

Eaton马尔可夫链、其共轭伙伴与P可容许性AI 解读

Suppose that X is a random variable with density $f(x|\theta)$ and that $\pi(\theta|x)$ is a proper posterior corresponding to an improper prior $\nu(\theta)$. The prior is called $\mathscr{P}$-admissible if the generalized Bayes estimator of every bounded function of $\theta$ is almost-$\nu$-admissible under squared error loss. Eaton showed that recurrence of the Markov chain with transition density $R(\eta|\theta) = \int \pi(\eta|x)f(x|\theta) dx$ is a sufficient condition…

自动化 Monte Carlo EM 算法最大化广义线性混合模型似然AI 解读

Summary Two new implementations of the EM algorithm are proposed for maximum likelihood fitting of generalized linear mixed models. Both methods use random (independent and identically distributed) sampling to construct Monte Carlo approximations at the E-step. One approach involves generating random samples from the exact conditional distribution of the random effects (given the data) by rejection sampling, using the marginal distribution as a candidate. The second method…

广义线性混合模型中的预测标准误AI 解读

The unconditional mean squared error of prediction (UMSEP) is widely used as a measure of prediction variance for inferences concerning linear combinations of fixed and random effects in the classical normal theory mixed model. But the UMSEP is inappropriate for generalized linear mixed models where the conditional variance of the random effects depends on the data. When the random effects describe variation between independent small domains and domain-specific prediction is…

包含空间协变量的鱼类物种丰富度分析AI 解读

Legislation passed in 1990 reducing the allowable sulfur dioxide emission levels in the United States is expected to reduce acidity in the Adirondack region of New York State. The number of fish species in a lake (species richness) depends on a number of physical and chemical factors, including the area, elevation, and acidity of the lake. Data on these and other factors are available for 1,166 Adirondack lakes. The data are analyzed with the goal of quantifying the effects…