Mathematical Sciences: Bayesian Inference and Computing

我们的研究是面向实现贝叶斯推理。最近人们对贝叶斯统计方法越来越感兴趣，部分原因是计算能力的进步使其在许多环境中变得可行，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以前的贝叶斯统计研究的基础上，其中一部分由NSF资助。我们的主要关切是：（1）审查和评估通过正式规则选择先验概率分布的方法，并进一步开发评估选择敏感性的方法;（2）研究贝叶斯假设检验的近似和精确计算方法;（3）修改和增强数值积分技术和后验分布的蒙特卡罗模拟;此外，统计计算环境的改进，包括使用动画和三维渲染的可视化不确定性在更高的维度;（4）进一步的工作的基础上的主观概率;（5）其他几个主题有关我们以前的工作启发先验和渐近近似。 在分析数据时，有效地将所有信息源联合收割机结合起来是很重要的。 贝叶斯统计方法是专门为这一目的而设计的。 我们的研究重点是寻找实用的方法来实现贝叶斯方法，并调查这些方法的理论基础。 我们关注的是计算和图形技术的发展，使贝叶斯推理在复杂的问题中是可行的。 这些包括：模拟，动画和统计计算环境的建设。 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.