Sparse Bayesian learning (SBL) is an advanced statistical modeling technique for inverse problems that builds upon traditional Bayesian methods by integrating hierarchical structures within prior distributions. This approach allows for extracting intricate relationships between parameters at various levels, fostering information sharing throughout the model. It is particularly effective when dealing with limited, noisy, or indirect data, yielding more accurate and robust inferences. Consequently, SBL has proven successful in diverse fields, such as machine learning, signal processing, remote sensing, and mathematical biology. In this talk, we introduce a novel approach for efficient Markov chain Monte Carlo (MCMC) sampling from the complex SBL posterior. 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. These maps transform the complex target posterior into a simpler reference distribution equipped with a standard normal prior that can be sampled more efficiently. Numerical experiments will demonstrate order-of-magnitude speedups for standard MCMC techniques. This talk is based on joint work with Youssef Marzouk (MIT).