Bayesian Information Criteria and Problems of Parameter Identifiability

本课题研究的是通过信息标准的优化来选择统计模型。具体而言，研究者开发了不规则模型选择问题的贝叶斯信息准则的推广，例如确定混合模型中的组件数量或潜在因素模型中的因素数量。这些不规则模型选择问题的主要困难是缺乏参数可辨识性。研究广泛应用的统计模型的可辨识性，为新信息准则的应用提供数学基础。实际上，每一个科学数据分析都会带来统计模型选择的问题，不同的统计模型捕捉不同的科学假设。在许多应用中，假设涉及不能或未观察到的潜在变量。例如，这些潜在变量可以是心理学研究中的智力概念，或者是医学研究中描述患者基因组成的变量。使用这些潜在变量制定的统计模型通常缺乏作为标准统计程序证明基础的规律性。该项目开发了用于模型选择的新统计技术，这些技术在理论上是合理的，并允许在具有影响的未观察变量的广泛应用中改进对模型不确定性的评估。

This project is concerned with statistical model selection by means of optimization of information criteria. Specifically, the investigator develops a generalization of the Bayesian information criterion for irregular model selection problems, such as determining the number of components in mixture models or the number of factors in latent factor models. The main difficulty in these irregular model selection problems is a lack of parameter identifiability. The investigator studies identifiability properties of widely used statistical models to provide the mathematical foundation for application of the new information criterion.Virtually every scientific data analysis brings about a problem of statistical model choice, where the different statistical models capture different scientific hypotheses. In many applications, the hypotheses involve latent variables that cannot or were not observed. Such latent variables could be, for instance, notions of intelligence in a psychological study or variables describing a patient's genetic composition in a medical study. Statistical models that are formulated using such latent variables typically lack the regularity properties that underlie the justification of standard statistical procedures. This project develops new statistical techniques for model selection that are theoretically justified and allow for an improved assessment of model uncertainty in a wide array of applications in which influential unobserved variables are at play.