Bayesian Modeling and Computations

我的研究计划的广泛目标是开发和实施复杂的统计模型，重点是贝叶斯设置下的计算。我试图研究与实际问题相对应的学科领域。在社交网络的主题领域，我正在研究与人际感知相对应的“准确性”评估。三元数据是最丰富的一种人际感知数据，研究中的所有个体对每一对个体都有评分。准确性评估是一个基本问题，也是社会和个人心理学中最古老的问题之一。人们对他人的看法是正确的吗？我正在考虑的方法扩展了传统的随机效应模型，从经典的角度来看，分析仍然是虚幻的。各种建模假设和先验分布的引入导致了复杂的高维贝叶斯模型。第二个项目涉及一种特殊类型的分类数据的建模和分析。在调查中，分类数据经常被错误分类。例如，“正确”分类为第一类的主题可能被错误地分类为第二类。在这种情况下，当忽略错误分类的影响时，可能会出现严重偏差估计。在这个项目中，我将尝试解释从单个多项队列扩展到特定主题协变量的数据的错误分类。此外，将考虑是否存在金标准数据。模型的复杂性将不可识别性问题带到了最前沿，并且需要引出主观先验分布。统计学在体育运动中的应用是我所涉及的一个普遍的应用领域。我打算研究的一个问题是20岁板球的最佳球队选择。这项工作的一个创新之处在于，传统的板球统计数据没有被用于确定球员的价值。相反，阵容中有和没有特定球员的跑动差异才是真正的价值衡量标准。我提出了一个经验贝叶斯程序来推断球员的击球和保龄球特征。通过模拟，这些特征可以用来评估给定阵容的质量。然后可以通过模拟退火算法的微调在一个巨大的组合空间上对阵容进行优化。这项工作可能有利于球队的阵容选择。我正在考虑的一个方法论项目涉及重要性抽样算法的发展。重要性抽样是一种基本的抽样策略，在贝叶斯统计中出现的积分逼近中很重要。重要性抽样与流行的马尔可夫链方法相比有两个特定的优点：(i)生成的变量是独立的，这简化了误差评估；（ii）不需要诊断收敛到平稳。然而，有时人们会说，重要性抽样在高维情况下效果不佳。我不认为这种观点完全正确。相反，对于高维问题，重要性抽样还没有使用足够丰富的族来实现。通常，多变量正态分布和学生分布已被用于重要性抽样。我的目标是开发多变量重要性采样算法，使用可选的多变量分布，如偏对称族。我还计划开发一个重要性采样的自适应组件，从而使重要性采样器在连续的迭代中得到改进。获得自适应算法收敛性的证明是项目的一部分。

The broad objective of my research proposal is the development and implementation of complex statistical models with a focus on computation in Bayesian settings. I attempt to work in subject areas corresponding to real problems.In the subject area of social networks, I am investigating the assessment of ``accuracy'' corresponding to interpersonal perceptions. Triadic data are the richest type of interpersonal perception data where all individuals in a study have ratings on every pair of individuals. The assessment of accuracy is a fundamental problem and is one of the oldest issues in social and personal psychology. Are people's perceptions of others valid? The approach that I am considering extends a traditional random effects model where analysis has remained illusive from a classical point of view. Various modeling assumptions and the introduction of prior distributions leads to complex and high dimensional Bayesian models.A second project involves the modeling and analysis of a special type of categorical data. In surveys, categorical data are often misclassified. For example, a subject whose ``true'' classification is the first category may be incorrectly classified in the second category. In such cases, severely biased estimators can occur when the effect of misclassification is ignored. In this project, I will attempt to account for misclassification for data extending from a single multinomial cohort to the case of subject-specific covariates. In addition, the presence of gold standard data will be considered. The complexity of the models brings nonidentifiability issues to the forefront and the need to elicit subjective prior distributions.Statistics in sport is a general application area in which I am involved. One problem that I plan to pursue concerns optimal team selection in Twenty20 cricket. An innovation in this work is that traditional cricket statistics are not used in determining player worth. Instead, run differential with and without a given player in the lineup is the true measure of value. I propose an empirical Bayes procedure to infer batting and bowling characteristics of players. Via simulation, these characteristics can be used to assess the quality of given lineups. Lineups may then be optimized over a huge combinatorial space via fine tuning of the simulated annealing algorithm. This work may benefit teams in terms of roster selection.A methodological project which I am considering concerns the development of importance sampling algorithms. Importance sampling is a fundamental sampling strategy that is important in the approximation of integrals arising in Bayesian statistics. Importance sampling has two specific advantages over popular Markov chain methods: (i) generated variates are independent which simplifies error assessment and (ii) there is no need to diagnose convergence to stationarity. Nevertheless, it is sometimes said that importance sampling does not work well in high dimensions. I do not believe this sentiment to be entirely true. Rather, importance sampling has not been implemented using sufficiently rich families for higher dimensional problems. Typically, the multivariate normal and Student distributions have been used for importance sampling. It is my goal to develop multivariate importance sampling algorithms using alternative multivariate distributions such as skew-symmetric families. I also plan on developing an adaptive component to importance sampling whereby the importance sampler is improved over successive iterations. Obtaining proofs of the convergence of the adaptive algorithm forms part of the project.