Monte Carlo Methods for Complex Problems: From Data Augmentation to Likelihood Free Inference

高度复杂的随机模型在科学应用中经常出现。它们经常导致分析上或计算上难以处理的似然函数的统计推断问题。这类问题超出了当前蒙特卡罗方法的极限。这一建议的目标是开发高效的蒙特卡罗算法来解决具有难以处理的似然的统计推断问题。拟议的研究考虑了两类没有可能性的问题。在第一类中，如果将问题放在适当的扩充空间中，则可能性以分析形式可用。针对这类问题，提出了一种新的数据扩充方案，从而得到了一种更有效的马尔可夫链蒙特卡罗算法。在第二类中，模型是一个“黑箱”，只有生成的随机机制可用于模拟模型中的数据。针对这类问题，提出了几种无似然蒙特卡罗算法，扩展了现有蒙特卡罗方法的能力。所提出的方法被应用于人口遗传学、面板研究和水文模型中的推理问题。所提出的研究满足了迫切需要开发创新的蒙特卡罗方法来解决具有难以处理的似然函数的问题。这在统计学上具有根本性的重要性。它使研究人员能够专注于科学上可信的统计模型，而不必担心数学上的困难。所提出的方法的应用包括分子群体遗传学中的统计推断，它可以帮助定位导致遗传病的基因，以及可以用来预测地下水流量的水文模型的贝叶斯校准。拟议的研究通过让研究生和本科生直接参与拟议的研究、将拟议的算法纳入相关课程以及向科学界传播研究成果，对教育产生重大影响。

Highly complex stochastic models arise frequently in scientific applications. They often lead to statistical inference problems with analytically or computationally intractable likelihood functions. Such problems lie beyond the limit of current Monte Carlo methods. The goal of this proposal is to develop efficient Monte Carlo algorithms for statistical inference problems with intractable likelihoods. Proposed research considers two categories of problems without likelihoods. In the first category, the likelihood is available in analytical forms if the problem is put into an appropriate augmented space. For such problems, a new data augmentation scheme is proposed which leads to a more efficient Markov chain Monte Carlo algorithm. In the second category, the model is a "black box" and only a generating stochastic mechanism is available to simulate data from the model. For such problems, several likelihood-free Monte Carlo algorithms are proposed which extend the power of current Monte Carlo methods. The proposed methods are applied to inference problems in population genetics, panel studies, and hydrological models.The proposed research addresses the urgent need to develop innovative Monte Carlo methodology for problems with intractable likelihood functions. This is of fundamental importance in statistics. It allows researchers to concentrate on scientifically plausible statistical models without worrying about mathematical intractability. Applications of the proposed methods include statistical inference in molecular population genetics which can help locating genes that are responsible for genetic diseases, and Bayesian calibration of hydrological models which can be used to predict ground-water flow. The proposed research has significant impact on education through involvement of graduate and undergraduate students directly in the proposed research, incorporation of proposed algorithms into related courses, and dissemination of research results to the scientific communities.