Optimal bayesian inference for high dimensional, graphical and functional data
Abstract
Advances in technology have resulted in massive datasets collected from all aspects of modern life such as internet search, mobile apps, social networking, cloud-computing, wearable devices, as well as from more traditional sources such as bar-code scanning, satellite imaging, air traffic control, banking and finance, and genomics. The complexity of such data warrants the use of flexible models involving many parameters. New challenges arise in the computation and statistical analysis regarding such data. Finding a lower-dimensional structure in the data is key to analyzing these complex datasets. Specifically, in regression problems with a large number of predictors, selection of relevant variables, making accurate inference and prediction and quantifying uncertainty are problems of interest. Learning the structure of a graph in a graphical model describing dependence among variables conditional on other variables is extremely important. Bayesian methods have the ability to naturally quantifying uncertainty in prediction and structure learning along with estimates, which is very desirable. However, Bayesian methods can be computationally intensive and their theoretical properties are less understood. Recently, Bayesian methods for finding lower-dimensional structures have seen a lot of interest, and their properties are being studied theoretically. Questions that arise are whether the posterior distribution contracts near the true value of the parameter at the minimax optimal rate, and whether the correct lower-dimensional structure is discovered with high posterior probability. Further of special interest is to know if a credible region constructed from the posterior distribution has adequate coverage in the frequentist sense. Results of such types can identify good Bayesian methods, potentially detect possible pitfalls and can also guide to appropriate choices of the prior distributions to avoid undesirable properties. However, realistic models are usually a lot more complex due to complicating factors such as the presence of measurement errors, missing observations, mixed-effects, heteroscedasticity and so on. If these nuisances are not properly incorporated in the analysis, the Bayesian procedure resulting from simplified incorrect modeling assumptions may lead to misleading conclusions. Bayesian methods that incorporate such structures need to be constructed and their convergence properties need to be investigated. In this proposed research, results on properties of posterior distributions will be extended to cover a wide variety of models useful in practical applications such as high dimensional linear models with added complications like measurement error, missing values, generalized linear models, graphical models with measurement errors, hidden Gaussian graphical models, exponential trace-class models, hub-and-spoke graphical models, multiple change-point models and recovery of functional signals varying over a graph. New technical tools, as well as computer packages for ready use, will be developed through the proposed research. Graduate students will be trained and will write doctoral theses. The methods will be applied to datasets potentially interesting for defense applications.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Jun 25, 2021
- Source ID
- W911NF2110239
Entities
People
- Subhashis Ghosal
Organizations
- Army Contracting Command
- North Carolina State University
- United States Army