Problems in Bayesian Model Selection and Development and Analysis of Markov Chain Sampling Algorithms

贝叶斯模型选择与马尔可夫链抽样算法开发中的问题及分析

• 批准号: 1106395
• 负责人: James Hobert
• 金额: $ 24万
• 依托单位: University of Florida
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1106395&HistoricalAwards=false
• 关键词: Problems; Bayesian; Model; Selection; Development

Bayesian methods are now routinely used in very complex models, with posterior distributions estimated by Markov chain Monte Carlo (MCMC) methods. There are two consequences to this. First, complex Bayesian models are virtually always governed by some hyperparameters, which have a large impact on subsequent inference. Therefore, there is now a strong need for methods that enable selection of these hyperparameters. Second, the Markov chains used to estimate the posterior distributions now run in non-standard spaces, for example large function spaces, and there is a need for the development of MCMC methods that will work well in non-standard spaces. The investigators develop methods for efficiently estimating marginal likelihoods for large number of hyperparameter values. This will enable implementation of the empirical Bayes method, and also enables users to determine classes of hyperparameter values which constitute reasonable choices. The exploration of intractable posterior distributions resulting from complex Bayesian models often requires MCMC. Unfortunately, in contrast with classical Monte Carlo, establishing central limit theorems (CLTs) for MCMC estimators is not straightforward. This is a serious practical problem because the ability to choose an appropriate MCMC sample size hinges upon the existence of a CLT. The investigators use spectral methods to develop checkable sufficient conditions for CLTs as well as methods for comparing the asymptotic efficiency of MCMC algorithms with the same target distribution. They apply the theoretical results to very concrete problems of model selection and assessment.Model selection in complex situations is an important and pervasive problem in scientific and medical research. It includes in particular variable selection in regression, where a few important variables are to be selected from many candidates and used for understanding, prediction and decision making. Different models can lead to different conclusions, with potential impact on public policy. The investigators develop efficient computational methods for determining optimal models in complex settings. The project has an educational component in that graduate students are involved in the research under the supervision of the investigators.

贝叶斯方法现在通常用于非常复杂的模型，后验分布估计马尔可夫链蒙特卡罗（MCMC）方法。 这会带来两个后果。 首先，复杂的贝叶斯模型几乎总是由一些超参数控制，这些超参数对后续的推理有很大的影响。 因此，现在非常需要能够选择这些超参数的方法。 其次，用于估计后验分布的马尔可夫链现在在非标准空间中运行，例如大型函数空间，并且需要开发在非标准空间中工作良好的MCMC方法。 研究人员开发了有效估计大量超参数值的边缘似然性的方法。 这将使经验贝叶斯方法的实施，也使用户能够确定超参数值的类别，构成合理的选择。 复杂的贝叶斯模型所产生的难以处理的后验分布的探索往往需要MCMC。 不幸的是，与经典的Monte Carlo相比，建立MCMC估计的中心极限定理（CLTs）并不简单。 这是一个严重的实际问题，因为选择适当MCMC样本量的能力取决于CLT的存在。 研究人员使用谱方法来开发CLT的可检查的充分条件，以及比较具有相同目标分布的MCMC算法的渐近效率的方法。 他们将理论结果应用于模型选择和评估的非常具体的问题。复杂情况下的模型选择是科学和医学研究中一个重要而普遍的问题。 它特别包括回归中的变量选择，其中要从许多候选变量中选择一些重要变量，并用于理解，预测和决策。 不同的模型可能会得出不同的结论，对公共政策产生潜在影响。 研究人员开发了有效的计算方法来确定复杂环境中的最佳模型。 该项目有一个教育部分，研究生在调查人员的监督下参与研究。