New algorithms for Bayesian Computation

贝叶斯统计学是一种基于观测数据了解未知参数和变量的高度原则性的方法。在该方法中，未知参数的现有知识由先验分布表示。一旦观察到新数据，则通过贝叶斯公式更新先验分布，以产生表示关于参数的更新知识的后验分布。关于感兴趣的参数的详细信息通常是通过计算推断方法从后验分布中获得的，例如马尔可夫链、蒙特卡罗或重要性抽样。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，例如当似然函数太昂贵而无法评估时，或者当统计模型作为没有显式似然的生成性模型给出并且只能用于模拟数据时。这个项目的目标是开发在这些情况下保持计算效率的贝叶斯推理方法。这一结果将使贝叶斯方法在许多科学和技术领域得到更广泛的使用。该项目还将通过研究生参与研究的表现来帮助他们进行培训。具体地说，该项目将创建新的计算工具来解决当前算法面临的两个问题，即如何从连续变量隐马尔可夫模型的后验分布中进行采样，以及如何在似然函数不可用的情况下设计从后验分布进行模拟的序贯方法。隐马尔可夫模型在工程和生物科学中有着广泛的应用，但目前该模型中的贝叶斯推理算法只有在所涉及的变量为离散变量时才可用。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。这个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的似然函数的模型中进行近似贝叶斯计算。这些模型可以产生于许多科学领域，如系统发育学和基于计算机的实验。目前只有一种算法(ABC算法)可用于此类模型的贝叶斯推理。通过开发一种可以极大地提高该算法的计算效率的扩展，该项目的研究将使上述科学领域受益。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.