The project aims to develop and test novel methodologies for efficient Bayesian computational statistics. Specifically, the project investigates novel approaches for efficient Markov chain Monte Carlo (MCMC) sampling from complex probability distributions in sparse Bayesian learning (SBL). The core innovation involves using prior-normalizing transport maps, which are deterministic couplings that transform the hierarchical sparsity-promoting SBL prior into a standard normal distribution. Numerically demonstrating the practical advantage of these methods requires computational experiments for various inverse problems—including high-dimensional signal deblurring and non-linear generative models. These experiments lead to high-dimensional and complicated posterior distributions that are computationally expensive to sample from.