Efficient Sampling for Sparse Bayesian Learning Using Hierarchical Prior Normalization

We introduce an approach for efficient Markov chain Monte Carlo (MCMC) sampling for challenging high-dimensional distributions in sparse Bayesian learning (SBL). The core innovation involves using hierarchical prior-normalizing transport maps ( TMs), which are deterministic couplings that transform the sparsity-promoting SBL prior into a standard normal one. We analytically derive these prior-normalizing TMs by leveraging the product-like form of SBL priors and Knothe–Rosenblatt rearrangements. These transform the complex target posterior into a simpler reference distribution equipped with a standard normal prior that can be sampled more efficiently. Specifically, one can leverage the standard normal prior by using more efficient, structure-exploiting samplers. Our numerical experiments on various inverse problems—including signal deblurring, inverting the nonlinear inviscid Burgers equation, and recovering an impulse image—demonstrate significant performance improvements for standard MCMC techniques.