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，并且它们在应用科学中的使用继续增长。这种流行程度的增加主要是由于马尔可夫链蒙特卡罗（MCMC）算法的可用性，该算法允许估计后验分布。然而，在大量的应用程序中，MCMC为基础的估计报告没有一个有效的措施，其质量和没有一致的策略，以决定何时停止模拟。在很大程度上，这是由于这样一个事实，即分析（渐近和非渐近）的通常MCMC估计通常是具有挑战性的。例如，与经典的蒙特卡罗方法相反，建立允许MCMC估计量的渐近分析的中心极限定理（CLT）并不简单。这是一个严重的实际问题，需要注意，因为目前有效的策略，评估质量的估计和选择（MCMC）样本量的理论假设，如CLTs的存在。本项目旨在解决这一问题，包括两个部分。第一部分是基于Gibbs抽样的MCMC估计的渐近和非渐近分析，用于几个广泛适用的贝叶斯版本的广义线性混合模型。调查人员认为概率单位和身份链接函数以及流行的选择适当和不适当的参数的先验密度。在该项目的第二部分中，调查人员试图推广两个变量吉布斯采样器的算法的收敛速度和它的连接的统计模型的参数化的几个结果。我们的目标是将已知的结果推广到一般的k-变量吉布斯采样器。由于吉布斯采样器是非常流行的MCMC算法，这些结果将可能有许多应用。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，这是应用科学中一个非常重要的问题。这一点如此重要的原因是，基于MCMC的误导性估计可能导致错误的结论，这可能会对公共政策产生负面影响。本研究的主要目标是找到简单的充分条件（用户可以检查）下，所考虑的MCMC程序是诚实的，也就是说，至少有一个有效的措施的质量估计和一个连贯的策略，决定何时停止模拟。更雄心勃勃的目标是为MCMC用户提供明确的（非渐近的）迭代次数的界限，以达到预定的精度水平。该项目中考虑的统计模型和MCMC算法在几乎每个科学学科中都有许多应用。因此，该项目产生的结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.