RI:Small:Matrix-structured statistical inference

许多跨越科学和工程的现代问题需要使用统计模型来描述潜在变量之间的概率关系。许多这些统计模型的一个核心组成部分是一个矩阵，它包括模型的参数。该建议特别关注一种称为图形模型的特殊子类，它使用加权图来表示底层变量上的分布。这种统计建模的主要用途是预测和推断：然而，对于一般模型来说，这些任务通常在计算上很困难或昂贵。一个重要的目标是轻松地执行这些推理任务(如果大致如此)。这项建议研究了与数值线性代数领域中看似无关的领域的最新进展建立联系并利用其来解决矩阵中的线性系统的方法，所述数值线性代数通过构建所谓的预条件矩阵来解决矩阵中的线性系统，所述预条件矩阵是线性系统矩阵的近似。这样的统计和图形模型被用于科学和工程问题：事实上，就连我们的手机也解决了图形模型推理问题，以解码它们接收的信号。因此，加速这些任务对于所有这些不同的应用程序都非常重要。研究人员参与了德克萨斯大学奥斯汀分校、计算工程与科学研究所以及统计和科学计算司的跨学科倡议；在这些倡议中，他们特别参与了跨学科和课程传播这种尖端研究。

Many modern problems across science and engineering require the use of statistical models that describe the probabilistic relationship among the underlying variable. A core component of many of these statistical models is a matrix, comprising the parameters of the model. This proposal focuses in particular on a special subclass called graphical models that use a weighted graph to represent a distribution over the underlying variables. The main use of such statistical modeling is prediction and inference: however these tasks are typically computationally intractable or expensive for general models. An important objective is to perform these inference tasks tractably if approximately. This proposal investigates approaches that make connections with and leverage recent advances in the seemingly unrelated field of numerical linear algebra that solve linear systems in matrices by building so-called preconditioner matrices that are approximations to the linear system matrices. Such statistical and graphical models are used across science and engineering problems: indeed even our cellphones solve graphical model inference problems to decode their received signals. Speeding up these tasks is thus of tremendous importance to all of these varied applications. The researchers are involved in interdisciplinary initiatives at the University of Texas, Austin; the Institute for Computational Engineering and Sciences, and the Division of Statistics and Scientific Computation; within which they are specially involved in disseminating such cutting edge research across disciplines and courses.