Monte Carlo methods for complex multimodal distributions with applications in Bayesian inference

当后验分布具有多个模式时，无条件期望（如后验均值）可能无法提供分布的信息摘要。出于这个问题，研究者提出开发马尔可夫链蒙特卡罗（MCMC）方法，可以产生足够的样本从域的吸引力的每个主要模式，因此构建估计的概率质量和条件期望给定的域。将开发计算方法，以建立基于MCMC样本的分布景观。本计画将贡献多模态后验分布下MCMC与贝氏推论的新方法，并推广自适应马尔可夫链的理论。基于多域采样器的框架，提出了一种新的算法，用于动态分组由低障碍分隔的域，并为分布构造子水平集树。该树包括作为终端节点的局部模式和作为内部节点的障碍。该项目还通过基于域的估计和算法开发贝叶斯推理方法，以量化后验模式及其吸引域的稳定性，并应用于贝叶斯缺失数据问题和结构估计。在双自适应MCMC框架下研究了全局移动多域采样器的收敛性和遍历性。一个理论模型，基于子水平集的树，将被开发，以促进MCMC算法的收敛性和效率分析。 许多学科中的科学问题可以通过从给定的概率分布中抽样来解决。蒙特卡罗方法，特别是马尔可夫链蒙特卡罗，是一类随机模拟算法，可以从几乎任何分布中抽取样本。然而，当分布具有多个局部模式时，这些算法的效率较低。因此，该项目的第一个意义在于它适用于各个科学领域的许多问题，包括统计物理，化学物理和计算生物学。另一方面，几乎没有现有的方法，可以提取有用的信息，多峰分布的Monte Carlo样本。拟议的项目包括系统地开发计算方法，用于通过分布景观的统一图形表示来构建关于多峰分布的新颖和全面的摘要。这可以极大地增强当前对统计和机器学习中许多问题的理解，例如，量化问题的难度并提供高维目标函数的可视化。

When a posterior distribution has multiple modes, unconditional expectations, such as the posterior mean, may not offer informative summaries of the distribution. Motivated by this problem, the investigator proposes to develop Markov chain Monte Carlo (MCMC) methods that may generate sufficient samples from the domain of attraction of every major mode and therefore construct estimates for the probability mass of and conditional expectations given a domain. Computational methods will be developed to build the landscape of a distribution based on an MCMC sample. This project will contribute novel methodologies on MCMC and Bayesian inference with multimodal posterior distributions, and generalize theory on adaptive Markov chains. A new algorithm, based on the framework of the multi-domain sampler, will be developed to group dynamically domains separated by low barriers and to construct the tree of sublevel sets for a distribution. The tree includes local modes as terminal nodes and barriers as internal nodes. This project also develops Bayesian inference methods via domain-based estimation and algorithms to quantify the stability of a posterior mode and its domain of attraction, with applications in Bayesian missing data problems and structure estimation. Convergence and ergodicity of the multi-domain sampler with global moves will be studied under the framework of doubly adaptive MCMC. A theoretical model, based on the tree of sublevel sets, will be developed to facilitate convergence and efficiency analysis of MCMC algorithms. Scientific problems in many disciplines may be solved by sampling from a given probability distribution. Monte Carlo methods, Markov chain Monte Carlo in particular, are a class of stochastic simulation algorithms that may draw samples from almost any distribution. However, these algorithms suffer from low efficiency when the distribution has multiple local modes. Therefore, the first significance of the proposed project comes from its applicability to many problems in various scientific fields, including statistical physics, chemical physics, and computational biology. On the other hand, there are almost no existing methods that can extract useful information about a multimodal distribution from Monte Carlo samples. The proposed project includes a systematic development of computational methods for constructing novel and comprehensive summaries about a multimodal distribution via a unified graphical representation for the landscape of a distribution. This can greatly enhance the current understanding of many problems in statistics and machine learning by, for example, quantifying the difficulty of a problem and providing visualization of a high-dimensional objective function.