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，例如后验均值，可能不提供关于该分布的信息性总结。受这个问题的启发，研究者建议发展马尔可夫链蒙特卡罗(MCMC)方法，该方法可以从每个主要模式的吸引域中产生足够的样本，从而构造给定域的概率质量和条件期望的估计。将开发基于MCMC样本的计算方法来构建分布的图景。该项目将为多模式后验分布的MCMC和贝叶斯推断提供新的方法，并推广自适应马尔可夫链的理论。基于多域采样器的框架，将开发一种新的算法来动态地对由低障碍分隔的域进行分组，并构建用于分布的子层集树。该树包括作为终端节点的本地模式和作为内部节点的障碍。该项目还开发了贝叶斯推理方法，通过基于域的估计和算法来量化后验模式及其吸引域的稳定性，并应用于贝叶斯丢失数据问题和结构估计。在双自适应MCMC框架下，研究具有全局运动的多域采样器的收敛和遍历性。为了便于MCMC算法的收敛和效率分析，将建立一个基于子层集树的理论模型。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方法，是一类可以从几乎任何分布中提取样本的随机模拟算法。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，提出的项目的第一个意义来自于它对包括统计物理、化学物理和计算生物学在内的各个科学领域的许多问题的适用性。另一方面，几乎没有现有的方法可以从蒙特卡罗样本中提取关于多峰分布的有用信息。拟议的项目包括系统地开发计算方法，以便通过对分配情况的统一图形表示来构建关于多式联运分配的新颖和全面的摘要。这可以极大地增强目前对统计学和机器学习中许多问题的理解，例如，通过量化问题的难度和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.