Improving the efficiency of Markov chain Monte Carlo

提高马尔可夫链蒙特卡罗的效率

• 批准号: 293260-2007
• 负责人: Yuen, WaiKong
• 金额: $ 1.17万
• 依托单位: Brock University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=363152
• 关键词: Improving; efficiency; Markov; chain; Monte

Markov chain Monte Carlo (MCMC) algorithms are widely used in the scientific community. They are used to generate approximate samples from complicated target distributions resulting from statistical and applied probability models. In particular, generic algorithms such as Metropolis and Langevin algorithms are very popular because of the ease of implementations. However, it is well known that choosing the best proposal distribution is not an easy task. Failing to do that can result in an inefficient algorithm and misleading results. My research program will focus on the theory and implementation of MCMC algorithms, typically on unbounded Euclidean state spaces, motivated by distributions such as those encountered in Bayesian inference. The three main objectives of my proposed research program are:  1. Bounding the convergence rates quantitatively. I shall generalize and combine available techniques through the application of decomposition theorems, from discrete state spaces, to general state spaces. This work will have direct value for theoreticians working in MCMC and will hopefully unify approaches for estimating and optimizing rates of convergence of algorithms;  2. Optimizing the convergence rates by proper scaling of the proposal distribution, when the dimension of the target density approaches infinity. I shall continue to focus on local MCMC algorithms, loosely referring to methods which have increment distributions centered at (or close to) the current state, and on proving weak convergence results for a wider class of target and proposal distributions. This work will extend and refine the scaling guidelines from available theory for all practitioners in the area;  3. Suggest modifications and study practical implementation issues on some existing sampling algorithms (e.g. MCMC, adaptive MCMC, etc.) widely used in the scientific literature. This includes stabilities of adaptations, convergence and efficiencies of these algorithms, and other problems motivated by practitioners.  This work, which is foundational, will enrich the theories of MCMC methods and related extensions.

马尔可夫链蒙特卡罗（MCMC）算法在科学界有着广泛的应用。它们用于从统计和应用概率模型产生的复杂目标分布中生成近似样本。特别是，通用算法，如大都会和朗之万算法是非常流行的，因为易于实现。然而，众所周知，选择最佳建议分布并不是一件容易的事情。如果不这样做，可能会导致效率低下的算法和误导性的结果。我的研究计划将侧重于MCMC算法的理论和实现，通常在无界欧几里得状态空间上，由贝叶斯推理中遇到的分布驱动。我提出的研究计划的三个主要目标是：1。收敛速度的定量界。我将通过分解定理的应用，从离散状态空间到一般状态空间，推广和联合收割机可用的技术。这项工作将有直接价值的理论工作在MCMC，并希望统一的方法估计和优化收敛速度的算法; 2。当目标密度的维数趋于无穷大时，通过适当缩放建议分布来优化收敛速度。我将继续专注于局部MCMC算法，松散地指的是增量分布集中在（或接近）当前状态的方法，并证明更广泛的目标和建议分布的弱收敛结果。这项工作将从现有的理论中扩展和完善该领域所有从业者的缩放指南; 3.对一些现有的采样算法（例如MCMC，自适应MCMC等）提出修改建议并研究实际实现问题。在科学文献中被广泛使用。这些问题包括算法的自适应稳定性、算法的收敛性和效率，以及其他由实际工作者提出的问题，这些工作是MCMC方法的基础，将丰富MCMC方法及其相关扩展的理论。