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

贝叶斯方法现在通常用于非常复杂的模型，其后验分布由马尔可夫链蒙特卡罗（MCMC）方法估计。这有两个后果。首先，复杂的贝叶斯模型实际上总是由一些超参数控制，这对后续推理有很大的影响。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，并且需要开发能够在非标准空间中良好工作的MCMC方法。研究人员开发了有效估计大量超参数值的边际似然的方法。这将使经验贝叶斯方法的实现成为可能，也使用户能够确定构成合理选择的超参数值的类别。探索由复杂贝叶斯模型产生的难以处理的后验分布通常需要MCMC。不幸的是，与经典蒙特卡罗相比，为MCMC估计量建立中心极限定理（clt）并不简单。这是一个严重的实际问题，因为选择合适的MCMC样本量的能力取决于CLT的存在。研究人员使用谱方法开发了clt的可检查充分条件，以及比较相同目标分布下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，从许多候选变量中选择几个重要变量，用于理解、预测和决策。不同的模型可以得出不同的结论，并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设置中的最佳模型。该项目具有教育成分，因为研究生在研究人员的监督下参与研究。

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.