Developing Conjugate Models for Exact MCMC free Bayesian Inference with Application to High-Dimensional Spatio-Temporal Data

The massive expansion in the production of data has led to natural computational challenges in uncertainty quantification. In particular, Bayesian methodology can account for sources of uncertainty, but requires techniques known to be computationally demanding. These difficulties are exacerbated when data are spatially and/or temporally correlated. The current solutions predominantly use either approximations or inefficient iterative methods such as Markov chain Monte Carlo (MCMC). This project resolves the computational challenges in uncertainty quantification with novel statistical methodology that does not require approximations and MCMC. Big data has impacted nearly every area of science, and as a result, methodological development and software for scalable, exact, MCMC free Bayesian methodology will have a substantial effect. Not only will the proposed methodology and software be an advancement in statistics, but it will be useful across a broad range of disciplines that deal with complex spatio-temporal processes such as neuroscience, climatology, demography, econometrics, ecology, meteorology, oceanography, and official statistics. The investigator will educate and train graduate students, and disseminate project findings through journal publications, public-use software, and conference presentations.

The objective of this project is to develop conjugate distribution theory for scalable Bayesian hierarchical models that create a larger framework for statisticians and subject matter scientist to perform MCMC free Bayesian inference without approximating the posterior distribution. In particular, this project will develop and extend the generalized conjugate multivariate (GCM) distribution, which allows one to simulate directly from the exact posterior distribution for a particular large class mixed effects models. This exact sample is referred to as Exact Posterior Regression (EPR). In Aim 1, the investigator will develop extensions of GCM and EPR to new settings including ordinal and nominal data, exact MCMC free inference for certain hyperparameters, and theoretical connections to existing statistical models. Aim 2 involves extensions of EPR to multivariate spatio-temporal and multiscale spatial data, allowing one to leverage several sources of dependence to improve predictions and perform spatial change of support (COS) without the use of MCMC or approximate Bayesian methods. To achieve scalability, in Aim 3, the investigator will develop an exact Bayesian hierarchical model that repeatedly subsets the data in an informative manner that does not impose additional assumptions on the data.

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.

Award Number: 2547531
Principal Investigator: Jonathan Bradley
Funds Obligated: $38,121
State: MO