New algorithms for Bayesian Computation

贝叶斯统计是一种基于观察数据学习未知参数和变量的高度原则的方法。在这种方法中，关于未知参数的现有知识由先验分布表示。一旦观察到新的数据，则通过贝叶斯公式更新先验分布，产生后验分布，该后验分布表示对参数的更新知识。关于感兴趣的参数的详细信息通常通过计算推理方法，如马尔可夫链蒙特卡罗或重要性抽样，从后验分布中获得。然而，这些计算推理方法在某些情况下可能会变得效率低下，例如当似然函数过于昂贵而无法评估时，或者当统计模型作为没有显式似然的生成模型并且只能用于模拟数据时。该项目的目标是开发在这些情况下仍然具有计算效率的贝叶斯推理方法。该结果将使贝叶斯方法在许多科学技术领域得到更广泛的应用。该项目还将通过研究生参与研究的表现，对研究生的培训作出贡献。具体来说，该项目将创建新的计算工具来解决当前算法面临的两个挑战，即如何从具有连续变量的隐马尔可夫模型的后验分布中采样，以及如何在没有似然函数的情况下从后验分布设计序列模拟方法。隐马尔可夫模型广泛应用于工程和生物科学中，但目前该模型中的贝叶斯推理算法仅在涉及的变量为离散变量时可用。通过为连续情况创建有效的算法，该项目的结果将使工程师和计算生物学家能够将这些模型应用于更广泛的问题。该项目的第二个目标是开发新的工具，用于在具有难以处理或未知似然函数的模型中进行近似贝叶斯计算。这些模型可以来自许多科学领域，如系统发育学和基于计算机的实验。目前对这类模型进行贝叶斯推理的算法只有一种（ABC算法）。通过开发一种可以大大提高该算法计算效率的扩展，本项目的研究将有益于上述科学领域。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Bayesian statistics is a highly principled approach to learn about unknown parameters and variables based on observed data. In this approach, existing knowledge about the unknown parameters is represented by the prior distribution. Once new data has been observed, then the prior distribution is updated by the Bayes formula to produce the posterior distribution which represents the updated knowledge about the parameter. Detailed information about the parameters of interest is usually obtained from the posterior distribution through computational inference methods such as Markov Chain Monte Carlo or Importance Sampling. However, these computational inference methods can become inefficient in some situations, such as when the likelihood function is too expensive to evaluate, or when the statistical model is given as a generative model without an explicit likelihood and can only be used to simulate the data. The goal of this project is to develop approaches to Bayesian inference that remain computationally efficient in these situations. The results will enable wider use of Bayesian methods in many areas of science and technology. The project will also contribute to the training of graduate students through their involvement in the performance of the research.Specifically, this project will create new computational tools to address two issues that are challenging for current algorithms, namely, how to sample from the posterior distribution in Hidden Markov Models with continuous variables, and how to design sequential methods for simulation from the posterior distribution even when the likelihood function is not available. Hidden Markov Models are widely used in the engineering and biological sciences, but currently algorithms for Bayesian inference in this model are available only if the variables involved are discrete variables. By creating efficient algorithms for the continuous case, the results of this project will enable engineers and computational biologists to apply these models to a much wider range of problems. The second goal of this project is to develop new tools for approximate Bayesian computation in models with intractable or unknown likelihood functions. These models can arise from many scientific areas such as phylogenetics and computer-based experiments. Currently there is only one available algorithm (the ABC algorithm) for Bayesian inference on this type of models. By developing an extension that can greatly improve the computational efficiency of this algorithm, the research in this project will benefit the aforementioned scientific areas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.