Analysis of Markov Chain Monte Carlo Algorithms with Applications to Bayesian Generalized Linear Mixed Models

贝叶斯统计方法已经变得流行起来，它们在应用科学中的使用也在继续增长。这种受欢迎程度的增加在很大程度上是由于马尔科夫链蒙特卡罗(MCMC)算法的可用性，该算法允许估计后验分布。然而，在大量的应用中，基于MCMC的估计被报告时没有有效的质量度量，并且没有一致的策略来决定何时停止模拟。在很大程度上，这是因为通常的MCMC估计量的分析(渐近和非渐近)通常是具有挑战性的。例如，与经典的蒙特卡罗方法不同，建立允许对MCMC估计量进行渐近分析的中心极限定理(CLT)并不简单。这是一个需要注意的严重的实际问题，因为目前评估估计质量和选择(MCMC)样本量的有效策略取决于理论假设，例如CLT的存在。这个项目解决了这个问题，由两部分组成。第一部分是广义线性混合模型基于Gibbs采样器的MCMC估计量的渐近和非渐近分析。研究人员考虑了概率和身份链接函数，以及参数的适当和不适当的先验密度的流行选择。在项目的第二部分，研究者试图推广关于双变量Gibbs采样器的几个结果，这些结果涉及算法的收敛速度及其与统计模型的参数化间的关系。目标是将已知结果推广到一般的k变量Gibbs采样器。由于Gibbs采样器是非常流行的MCMC算法，这些结果可能会有很多应用。特别是，它们可用于项目第一部分进行的分析。这个项目解决了贝叶斯统计中使用的MCMC程序中的估计质量，这是应用科学中的一个非常重要的问题。这一点之所以如此重要，是因为误导基于MCMC的估计可能会导致不正确的结论，这可能会对公共政策产生潜在的负面影响。这项研究的主要目标是找到简单、充分的条件(用户可以检查)，在这些条件下，所考虑的MCMC过程是诚实的；即，至少存在一种有效的估计质量度量和用于决定何时停止模拟的一致策略。更雄心勃勃的目标是为MCMC用户提供达到预定精度所需迭代次数的显式(非渐近)界限。这个项目中考虑的统计模型和MCMC算法在几乎所有的科学学科中都有大量的应用。因此，该项目产生的成果将被许多不同领域的研究人员使用。

Bayesian statistical methods have become popular and their use continues to grow in the applied sciences. This increase in popularity is largely due to the availability of Markov chain Monte Carlo (MCMC) algorithms which allow for the estimation of posterior distributions. However, in a large number of applications, MCMC-based estimates are reported without a valid measure of their quality and there is no coherent strategy for deciding when to stop the simulation. To a large extent, this is due to the fact that analyses (asymptotic and non-asymptotic) of the usual MCMC estimators are typically challenging. For example, as opposed to classical Monte Carlo methods, establishing the central limit theorems (CLTs) that allow for an asymptotic analysis of MCMC estimators is not straightforward. This is a serious practical problem that needs attention because the current valid strategies for assessing the quality of estimation and the choice of (MCMC) sample size rest upon theoretical assumptions such as the existence of CLTs. This project addresses this issue and consists of two parts. The first part consists of asymptotic and non-asymptotic analyses of MCMC estimators based on Gibbs samplers for several widely applicable Bayesian versions of the generalized linear mixed model. The investigator considers probit and identity link functions as well as popular choices of proper and improper prior densities for the parameters. In the second part of the project, the investigator attempts to generalize several results for two-variable Gibbs samplers concerning the convergence rate of the algorithm and its connection to the parametrization of the statistical model. The goal is to generalize known results to the general k-variable Gibbs sampler. Since Gibbs samplers are very popular MCMC algorithms, these results will likely have many applications. In particular, they could be used in the analyses performed in the first part of the project. This project addresses the quality of estimation in MCMC procedures used in Bayesian statistics, which is a very important issue in the applied sciences. The reason why this is so important is that misleading MCMC-based estimates can lead to incorrect conclusions, which could potentially negatively affect public policy. The main goal of this research is to find simple sufficient conditions (that the user can check) under which the considered MCMC procedures are honest; that is, there is at least one valid measure of the quality of estimation and a coherent strategy for deciding when to stop the simulation. The more ambitious goal is to provide the MCMC user with explicit (non-asymptotic) bounds on the number of iterations needed to achieve a predetermined level of accuracy. The statistical models and MCMC algorithms considered in this project have numerous applications in nearly every scientific discipline. Consequently, the results produced in this project will be used by researchers in many different fields.