Wasserstein guided nonparametric Bayes

Stochastic generative models are a cornerstone of applied statistical modeling and inference. A generative model is an abstraction, and often a simplification, of a data generating mechanism using probabilistic tools, where specific features of interest regarding the generating mechanism are encapsulated into parameters of the generative model. Bayesian statistical inference is a popular statistical paradigm for combining such generative models for data with prior information about model parameters in a principled fashion to perform statistical inference on the unknown parameters.

Some of the salient aspects behind the tremendous growth in popularity of Bayesian inference include principled incorporation of domain information, an in-built penalty for model complexity allowing automatic model selection, and facilitating borrowing of information across different domains via hierarchical modeling. However, being inherently model-based, Bayesian statistics is intrinsically susceptible to departures from the postulated generative model.

Through this project, the investigators will explore and develop new statistical methodology for performing Bayesian inference allowing flexible departures from the generative model under consideration. A major focus will be the user-friendliness of the proposed approaches, circumventing the need for a user to explicitly build probabilistic models of increasing richness.

The research will be disseminated through articles at prominent avenues and research presentations. Additionally, software packages for the methods developed will be made available publicly. The investigators are committed to enhancing the pedagogical component of the proposal through advising students and developing graduate and undergraduate topic courses.

Flexible nonparametric Bayesian methods have gained in popularity to address perceived issues of traditional Bayesian modeling regarding model-misspecification. The last thirty years have seen a proliferation of such methods, both in mainstream statistics as well as the machine learning community, as we continue to encounter increasing levels of complexities in modern datasets.

However, nonparametric Bayesian methods can be challenging to implement as well as interpret. Furthermore, in many applications, the targets of interest are quite simple and it is essentially futile to model all aspects of the data. The fundamental aim of the proposed research is to develop a flexible Bayesian non-parametric approach that retains the generative modeling aspect of traditional parametric Bayesian modeling while avoiding a complete probabilistic specification of the data generating mechanism as typically performed in nonparametric Bayesian modeling.

This will be performed by defining a modified likelihood function, leveraging ideas from the empirical likelihood literature as well as optimal transport theory, that centers around a user-specified parametric family of densities. An automated calibration procedure will be developed to control the extent of centering around the parametric model. The investigators will offer a firm theoretical underpinning of the proposed procedure and develop computationally efficient algorithms to carry out inferential tasks.

The developed methods will be applied to scientific learning problems in neuroscience and nuclear physics to allow departures from existing scientific models in situations where their operating characteristics are less understood.

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.