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，主要是因为马尔可夫链蒙特卡罗方法可以估计后验分布。 沿着这种使用的增加，研究人员现在经常考虑更复杂的模型，例如具有许多水平的分层模型和具有许多潜在预测因子的回归模型。 考虑更复杂的模型有两个结果。 由于参数存在于更大的空间中，因此更需要开发MCMC方法，以准确估计给定模型的相关后验分布和期望值，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。 他们考虑的情况下，有一组模型索引的几个连续的超参数。 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套计算效率高的计划，用于估计一个参数的函数的后验期望作为前不断变化。 这使用户能够确定先验的哪些方面对后验的影响最大。 在复杂环境中使用的马尔可夫链通常涉及旨在加速收敛的增强。 但理论上对这些增强的效果知之甚少。 在这个项目中，从算子理论的技术被用来分析马尔可夫链的长期行为。 算子理论框架提供了更好地理解这些链的行为的工具，这种理解使得能够开发关于这些链产生的估计的准确性的结果，并且还提出了改进这些链或设计更好的链的方法。 研究者将算子理论的理论结果应用于模型选择和评估的具体问题，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。 而对于频率论方法来说，有大量的材料可以用来做诊断，对于贝叶斯方法来说，现有的方法要有限得多。 贝叶斯模型评估和敏感性分析的工具，以及将在本项目中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从探索非常大的空间的马尔可夫链产生的估计的准确性，并将使他们能够正确确定需要运行多长的链，以提供所需的准确性水平。