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

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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.