Development and Analysis of MCMC Algorithms and Computational Methods in Bayesian Sensitivity Analysis

贝叶斯敏感性分析中MCMC算法和计算方法的开发与分析

• 批准号: 0805860
• 负责人: James Hobert
• 金额: --
• 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: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2012-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805860&HistoricalAwards=false
• 关键词: Development; Analysis; MCMC; Algorithms; Computational

The use of Bayesian statistics in the applied sciences has increased dramatically over the last decade, largely because of the availability of Markov chain Monte Carlo methods to estimate the posterior distributions. Along with this increased use, researchers are now routinely considering more complex models, for example hierarchical models with many levels and regression models with many potential predictors. Consideration of more sophisticated models has two consequences. Because the parameters live in larger spaces, there is a stronger need for the development of MCMC methods that give accurate estimates of the relevant posterior distributions and expectations for a given model, and there is a stronger need for methods for doing model diagnostics and selection. An important component of this project is the development of tools for model assessment and sensitivity analysis.The investigators develop methods for efficiently calculating a very large number of Bayes factors, and plotting them. They consider situations in which there is a set of models indexed by several continuous hyperparameters. Calculation of the Bayes factors helps determine whether a subset of hyperparameter values constitutes a class of reasonable choices. The investigators also develop a set of computationally efficient schemes for estimating the posterior expectation of a function of a parameter as the prior is varied continuously. This enables users to determine which aspects of the prior have the biggest impact on the posterior. Markov chains used in complex settings often involve enhancements designed to speed up convergence. But little is known theoretically regarding the effect of these enhancements. In this project techniques from operator theory are used to analyze the long-term behavior of Markov chains. The operator theory framework provides tools to better understand the behavior of these chains and this understanding enables the development of results regarding the accuracy of estimates produced by these chains and also suggests ways to improve these chains or design better ones. The investigators apply the theoretical results obtained from operator theory 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. Whereas for frequentist methods there is available an extensive body of material for doing diagnostics, for Bayesian methods the methods that exist are much more limited. The tools for Bayesian model assessment and sensitivity analysis, together with the theoretical results regarding Markov chains that will be obtained in this project, will enable researchers to correctly evaluate the accuracy of estimates produced from Markov chains that explore very large spaces, and will enable them to correctly determine how long chains need to be run in order to provide a required level of accuracy.

在过去的十年中，贝叶斯统计在应用科学中的使用急剧增加，这在很大程度上是因为马尔可夫链蒙特卡罗方法可用于估计后验分布。随着这种使用的增加，研究人员现在经常考虑更复杂的模型，例如具有多个级别的分层模型和具有许多潜在预测因素的回归模型。考虑更复杂的模型有两个后果。由于参数存在于更大的空间中，因此更需要开发MCMC方法来准确估计给定模型的相关后验分布和预期，并且更需要进行模型诊断和选择的方法。该项目的一个重要组成部分是开发用于模型评估和敏感性分析的工具。研究人员开发了有效计算大量贝叶斯因子并绘制它们的方法。他们考虑有一组由几个连续的超参数索引的模型的情况。贝叶斯因子的计算有助于确定超参数值的子集是否构成一类合理的选择。研究人员还开发了一组计算高效的方案，用于估计参数函数的后验期望，因为先验信息是连续变化的。这使用户能够确定之前的哪些方面对后方的影响最大。在复杂环境中使用的马尔可夫链通常涉及旨在加快收敛的增强。但从理论上讲，人们对这些增强的影响知之甚少。在这个项目中，我们使用了算子理论中的技术来分析马尔可夫链的长期行为。算符理论框架提供了更好地理解这些链的行为的工具，这种理解使关于这些链产生的估计的准确性的结果得以发展，并还建议了改进这些链或设计更好的链的方法。研究人员将算子理论的理论成果应用于非常具体的模型选择和评估问题，在科学和医学研究中，复杂情况下的模型选择是一个重要而普遍的问题。它特别包括回归中的变量选择，其中一些重要的变量是从许多候选变量中选择出来的，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，这可能会对公共政策产生影响。然而，对于频率法，有大量的材料可用于进行诊断，而对于贝叶斯方法，现有的方法要有限得多。贝叶斯模型评估和敏感性分析工具，以及本项目中将获得的关于马尔可夫链的理论结果，将使研究人员能够正确评估探索非常大空间的马尔可夫链产生的估计的准确性，并使他们能够正确确定需要运行多长链才能提供所需的精度水平。