Computational Methods for Structured Data and Models

许多领域的科学家，包括遗传学、神经科学、生态学和经济学，正在获得比以往更复杂过程的更丰富的测量方法。这为回答以前无法触及的科学问题提供了令人兴奋的机会。例如，在许多地点长期收集的疾病流行率测量值为估计疾病的传播提供了机会。潜变量模型是将观察到的数据建模为未观察到的潜在随机变量的简单变换，是一种从复杂过程的测量中提取科学问题答案的流行方法。它们是灵活的，但它们也可能在计算上令人望而却步。因此，使用潜在变量模型的科学家可能不得不满足于未知质量的近似值或特定的简化，这些简化提供了较差的估计或无法回答感兴趣的问题。本项目旨在开发新的方法来拟合潜在变量模型，使其计算效率更高，更可靠，更易于访问。参与该项目将培训各级统计人员，重点是来自统计研究中代表性不足的人口的统计人员。具体来说，研究者将监督研究生参与研究，指导本科生开发引人入胜的推广材料，参与当地高中的推广活动，并为早期职业教师领导写作小组。当拟合潜在变量模型时，会出现各种计算挑战。即使已知表征潜在变量模型的少量参数，也很难从给定观测数据的潜在变量的条件分布中进行表征和模拟。此外，估计潜在变量模型的未知参数可能很困难，因为数据的可能性对应于一个高维积分，对于这个积分，可能无法获得封闭形式的表达式，或者计算起来很昂贵。即使可行的方法是可用的，从业者也很难在没有开源软件和详细教程的情况下实现潜在变量模型。因此，本项目旨在提供(i)从高维潜在变量的条件分布进行模拟的新方法，（ii）潜在变量模型参数的最大似然估计的改进方法，以及（iii）允许从业者实施它们的通用统计软件。关于(i)， PI计划开发新的路径方法，用于从给定数据的高维潜在变量的条件分布进行模拟，利用目标条件分布与相关或近似分布的关系。关于（ii）， PI将开发利用第一个目标中引入的路径模拟方法的潜在变量模型参数的最大似然估计的改进方法。关于（三），PI将把新方法应用于疾病制图和全基因组关联研究，并开发R包，使其他从业者能够实施这些方法。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Scientists in many fields, including genetics, neuroscience, ecology, and economics, are obtaining richer measurements of more complex processes than ever before. This offers thrilling opportunities to answer scientific questions that were previously out of reach. For instance, measurements of disease prevalence collected at many locations over time provide the opportunity to estimate the spread of disease. Latent variable models, which model the observed data as a simple transformation of unobserved latent random variables, are a popular approach to extracting answers to scientific questions from measurements of complex processes. They are flexible, but they can also be computationally prohibitive. As a result, scientists using latent variable models may have to settle for approximations of unknown quality or ad-hoc simplifications that provide poor estimates or fail to answer questions of interest. This project aims to develop novel methods for fitting latent variable models that are more computationally efficient, reliable, and accessible. Involvement in the project will train statisticians at all levels, with a focus on statisticians from populations that are underrepresented in statistics research. Specifically, the investigator will supervise graduate student involvement in the research, guide the development of engaging outreach materials by undergraduate students, participate in outreach at local high schools, and lead writing groups for early-career faculty. A variety of computational challenges arise when fitting latent variable models. It can be difficult to characterize and simulate from the conditional distribution of the latent variables given observed data, even when the small number of parameters characterizing the latent variable model are known. Furthermore, it can be difficult to estimate the unknown parameters of a latent variable model because the likelihood of the data corresponds to a high dimensional integral for which a closed-form expression may be unavailable or expensive to evaluate. Even when feasible methods are available, it can be difficult for practitioners to implement latent variable models without access to open-source software and detailed tutorials. Accordingly, this project aims to contribute (i) novel methods for simulating from the conditional distributions of high dimensional latent variables, (ii) improved methods for maximum likelihood estimation of latent variable model parameters, and (iii) versatile statistical software that allows practitioners to implement them. Regarding (i), the PI plans to develop novel pathwise methods for simulating from the conditional distributions of high dimensional latent variables given data that leverage the relationship of the target conditional distribution to related or approximate distributions. Regarding (ii), the PI will develop improved methods for maximum likelihood estimation of latent variable model parameters that leverage the pathwise simulation methods introduced in the first aim. Regarding (iii), the PI will apply the new methods to disease mapping and genome-wide association studies and develop R packages that allow other practitioners to implement the methods.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.