Empirical Bayes Gibbs sampling

The wide applicability of Gibbs sampling has increased the use of more complex and multi-level hierarchical models. To use these models entails dealing with hyperparameters in the deeper levels of a hierarchy. There are three typical methods for dealing with these hyperparameters: specify them, estimate them, or use a 'flat' prior. Each of these strategies has its own associated problems. In this paper, using an empirical Bayes approach, we show how the hyperparameters can be estimated in a way that is both computationally feasible and statistically valid.     

A Bayes factor approach with informative prior for rare genetic variant analysis from next generation sequencing data

The discovery of rare genetic variants through next generation sequencing is a very challenging issue in the field of human genetics. We propose a novel region‐based statistical approach based on a Bayes Factor (BF) to assess evidence of association between a set of rare variants (RVs) located on the same genomic region and a disease outcome in the context of case‐control design. Marginal likelihoods are computed under the null and alternative hypotheses assuming a binomial distribution for the RV count in the region and a beta or mixture of Dirac and beta prior distribution for the probability of RV. We derive the theoretical null distribution of the BF under our prior setting and show that a Bayesian control of the false Discovery Rate can be obtained for genome‐wide inference. Informative priors are introduced using prior evidence of association from a Kolmogorov‐Smirnov test statistic. We use our simulation program, sim1000G, to generate RV data similar to the 1000 genomes sequencing project. Our simulation studies showed that the new BF statistic outperforms standard methods (SKAT, SKAT‐O, Burden test) in case‐control studies with moderate sample sizes and is equivalent to them under large sample size scenarios. Our real data application to a lung cancer case‐control study found enrichment for RVs in known and novel cancer genes. It also suggests that using the BF with informative prior improves the overall gene discovery compared to the BF with noninformative prior.     

Bayesian predictive information criterion for the evaluation of hierarchical Bayesian and empirical Bayes models

The problem of evaluating the goodness of the predictive distributionsof hierarchical Bayesian and empirical Bayes models is investigated.A Bayesian predictive information criterion is proposed as anestimator of the posterior mean of the expected loglikelihoodof the predictive distribution when the specified family ofprobability distributions does not contain the true distribution.The proposed criterion is developed by correcting the asymptoticbias of the posterior mean of the loglikelihood as an estimatorof its expected loglikelihood. In the evaluation of hierarchicalBayesian models with random effects, regardless of our parametricfocus, the proposed criterion considers the bias correctionof the posterior mean of the marginal loglikelihood becauseit requires a consistent parameter estimator. The use of thebootstrap in model evaluation is also discussed.     

Hierarchical Bayes models for cDNA microarray gene expression

cDNA microarrays are used in many contexts to compare mRNA levels between samples of cells. Microarray experiments typically give us expression measurements on 1000-20 000 genes, but with few replicates for each gene. Traditional methods using means and standard deviations to detect differential expression are not satisfactory in this context. A handful of alternative statistics have been developed, including several empirical Bayes methods. In the present paper we present two full hierarchical Bayes models for detecting gene expression, of which one (D) describes our microarray data very well. We also compare the full Bayes and empirical Bayes approaches with respect to model assumptions, false discovery rates and computer running time. The proposed models are compared to existing empirical Bayes models in a simulation study and for a set of data (Yuen et al., 2002), where 27 genes have been categorized by quantitative real-time PCR. It turns out that the existing empirical Bayes methods have at least as good performance as the full Bayes ones.     

朴素贝叶斯判别的判别效用分析

通过模拟实验研究了朴素贝叶斯判别的独立性假定对误判率的影响.实验结果表明:对于两个二维正态总体,当两类的距离为1、2、3时,在真实密度下进行判别的误判率随类内特征间相关系数的增加而减小,而朴素贝叶斯判别的误判率与特征间相关系数变化无关,因此随着类内特征间相关系数的增加,朴素贝叶斯判别的误判率明显增加,当类内特征间相关系数均为0.8时,朴素贝叶斯判别的误判率分别提高43.8%、220.8%和785.7%;对于高维正态总体模拟实验显示出同样的规律.     

Empirical Bayes estimation of gene-specific effects in micro-array research

Micro-array technology allows investigators the opportunity to measure expression levels of thousands of genes simultaneously. However, investigators are also faced with the challenge of simultaneous estimation of gene expression differences for thousands of genes with very small sample sizes. Traditional estimators of differences between treatment means (ordinary least squares estimators or OLS) are not the best estimators if interest is in estimation of gene expression differences for an ensemble of genes. In the case that gene expression differences are regarded as exchangeable samples from a common population, estimators are available that result in much smaller average mean-square error across the population of gene expression difference estimates. We have simulated the application of such an estimator, namely an empirical Bayes (EB) estimator of random effects in a hierarchical linear model (normal-normal). Simulation results revealed mean-square error as low as 0.05 times the mean-square error of OLS estimators (i.e., the difference between treatment means). We applied the analysis to an example dataset as a demonstration of the shrinkage of EB estimators and of the reduction in mean-square error, i.e., increase in precision, associated with EB estimators in this analysis. The method described here is available in software that is available at .