Mathematical Sciences: Bayesian Inference and Computing

我们的研究方向是贝叶斯推理的实现。最近，人们对贝叶斯统计方法的兴趣越来越大，部分原因是计算能力的进步使其在许多情况下都是可行的，部分原因是贝叶斯数据分析可以利用来自其他来源的信息。我们的工作将建立在我们之前在贝叶斯统计方面的研究基础上，其中一部分研究已经得到了美国国家科学基金会的资助。我们的主要关注点是：(1)回顾和评估通过形式规则选择先验概率分布的方法，并进一步发展评估选择敏感性的方法；(2)贝叶斯假设检验的近似和精确计算方法的研究；(3)改进和改进了数值积分技术和蒙特卡罗后验分布模拟；此外，统计计算环境的改进，包括使用动画和三维渲染来可视化高维的不确定性；(4)进一步研究主观概率的基础；(5)与我们之前关于先验和渐近逼近的推导工作相关的其他几个主题。在分析数据时，有效地结合所有信息来源是很重要的。贝叶斯统计方法就是为这个目的量身定做的。我们的研究重点是寻找实现贝叶斯方法的实际方法，并研究这些方法的理论基础。我们关注计算和图形技术的发展，使贝叶斯推理在复杂问题中可行。其中包括：仿真、动画和统计计算环境的构建。我们还将研究支持贝叶斯技术的理论问题。这些问题包括主观概率的基础和数学近似的发展。

Our research is oriented toward implementation of Bayesian inference. There has been increasing interest recently in the Bayesian approach to statistics, in part because advances in computational ability have made it feasible in many settings, and in part because Bayesian analysis of data can make use of information from additional sources. Our work will build on our previous research in Bayesian statistics, part of which has been funded by NSF. Our main concerns are: (1) review and assessment of methods for choosing prior probability distributions by formal rules, and further development of methods for assessing sensitivity to the choices; (2) investigation of approximate and exact computational methods for Bayesian hypotheses testing; (3) modification and enhancement of numerical integration techniques and Monte Carlo simulation of posterior distributions; also, improvement of statistical computing environments including use of animation and three dimensional rendering for visualization of uncertainty in higher dimensions; (4) further work on the foundations of subjective probability; and (5) several other topics related to our previous work on elicitation of priors and asymptotic approximations. When analyzing data, it is important to combine all sources of information effectively. Bayesian statistical methods are tailored to this purpose. Our research focuses on finding practical ways to implement Bayesian methods and on investigating the theoretical basis for these methods. We are concerned with the development of computational and graphical techniques that make Bayesian inference feasible in complicated problems. These include: simulation, animation and the construction of statistical computing environments. We will also investigate theoretical issues that support Bayesian techniques. These issues include the foundations of subjective probability and the development of mathematical approximations.