Collaborative Research: New Developments in Direct Probabilistic Inference on Interest Parameters

统计学习的贝叶斯方法依赖于所有可观测和未知的概率模型。对问题的所有方面进行建模的需要可能会限制应用程序的范围，更广泛地说，可能会给数据分析人员带来负担，他们通常只对未知的某些特性感兴趣。该项目将开发一个数学严谨和计算高效的框架，在该框架中，贝叶斯学习可以仅根据感兴趣的特征直接进行。这减少了数据分析师的建模和计算负担，并提供了更广泛的关于贝叶斯学习的新见解。贝叶斯方法是一种强大而严格的统计学习框架。缺点是它需要一个完整的模型来描述可观测数据以及所有未知量，这可能会给数据分析人员带来负担。除了先前规范所面临的常见挑战之外，还存在错误规范偏差的风险。一个更微妙的复杂性是由于考虑了几个候选模型而产生的选择效果。在只有未知数中的一个特征是感兴趣的情况下，即，当存在感兴趣的参数和(可能高维的)滋扰参数并且只需要对前者进行推断时，数据分析师的负担被进一步夸大。也就是说，贝叶斯方法仍然要求数据分析师做出非常重要的努力来指定先验分布，并执行仅与滋扰参数相关的后验计算，这可以被视为一种浪费。然而，能够利用后验分布来推断兴趣参数仍然是一个理想的特征，所提出的研究旨在开发一种新的框架，用于直接对兴趣参数进行后验推断。这些直接后验(DIP)有效地瞄准了兴趣参数，使数据分析师有机会避免涉及讨厌参数的看似浪费的建模和计算工作。该项目将为有限和无限维兴趣参数构建DIP，具有严格的理论保证，并将开发高效的计算工具来促进基于DIP的推理。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The Bayesian approach to statistical learning relies on probabilistic models for all observables and unknowns. The need to model all aspects of the problem can restrict the scope of applications and, more generally, can be a burden to the data analyst who is often only interested in certain features of the unknowns. This project will develop a mathematically rigorous and computationally efficient framework in which Bayesian learning can be carried out directly in terms of only the features of interest. This reduces the modeling and computational burden on the data analyst and provides new insights about Bayesian learning more generallyA Bayesian approach is a powerful and rigorous framework for statistical learning. The downside is that it requires a full model for the observables as well as all unknown quantities, the specification of which can be a burden on the data analyst. In addition to the familiar challenges of prior specification, there are also risks of misspecification biases. A more subtle complication is due to selection effects that result from considering several candidate models. The data analyst's burden is further exaggerated in situations where only a feature of the unknowns is of interest, i.e., when there is an interest parameter and a (potentially high-dimensional) nuisance parameter and inference is required only for the former. That is, the Bayesian approach still requires that the data analyst make non-trivial efforts to specify prior distributions and carry out posterior computations relevant only to the nuisance parameter, which can be viewed as a waste. Yet having access to a posterior distribution for inference on the interest parameter is still a desirable feature, and the proposed research aims to develop a new framework for posterior inference directly on interest parameters. These direct posteriors (DiPs) effectively target the interest parameter, giving data analysts an opportunity to avoid the seemingly wasteful modeling and computation efforts involving nuisance parameters. This project will construct DiPs for finite- and infinite-dimensional interest parameters with rigorous theoretical guarantees, and will also develop efficient computational tools to facilitate DiP-based inference.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.