Over the past few years, there has been a noticeable increase in the amount of available data with complex dependent structure. Bayesian statistics is an approach to inference based on the Bayes’ theorem, which is interpretable and provides uncertainty quantification. These advantages have made Bayesian methods widely used across various applied fields, including social sciences, ecology, genetics, medicine and more. In this thesis, we advance the application of Bayesian methods for three different types of dependent data.

For the first project, we develop a Bayesian median autoregressive model for time series forecasting. This model utilizes time-varying quantile regression at the median, which inherits the robustness of median regression in contrast to the widely used mean-based methods. We use Bayesian model averaging to account for model uncertainty including the uncertainty in the autoregressive order, in addition to a Bayesian model selection approach.

The second project addresses image-on-scaler regression. We consider a Bayesian hierarchical Gaussian process model for image smoothing, that uses a flexible Inverse-Wishart process prior to handle within-image dependency. We propose a general global-local spatial selection prior that achieves simultaneous global (i.e., at the covariate-level) and local (i.e., at the pixel/voxel-level) selection. We introduce participation rate parameters that measure the probability for individual covariates to affect the observed images. This along with a hard-thresholding strategy leads to dependency between selections at the two levels, introduces extra sparsity at the local level, and allows the global selection to be informed by the local selection, all in a model-based manner.

The last project is on Gaussian graphical regression models with covariates. We use a tensor representation of the regression coefficients to describe the multi-level selection action achieved by the proposed prior: covariate-level, edge-level and local-level. Simultaneous multi-level selection is done by nesting a global-local spike-and-slab prior in a sparse group selection prior. This nested prior first achieves a global-level selection, excluding a covariate, by measuring the probability of the covariate being influential, and then, conditional on the outcome, performs edge-level selection in the manner of conventional Gaussian graphical regression models.

In a fully Bayes approach, we design Markov Chain Monte Carlo (MCMC) samplers for all three models and show in simulations and real data applications (with U.S. macroeconomic data, Autism brain imaging data, and human gene expression data respectively) that the proposed Bayesian methods are competitive with respect to existing models. Furthermore, the proposed Bayesian methods are also highly interpretable and able to provide joint uncertainty quantification via posterior samples for prediction and/or inference.