  Sparse Bayesian learning (SBL) has been extensively utilized in data-driven
modeling to combat the issue of overfitting. While SBL excels in
linear-in-parameter models, its direct applicability is limited in models where
observations possess nonlinear relationships with unknown parameters. Recently,
a semi-analytical Bayesian framework known as nonlinear sparse Bayesian
learning (NSBL) was introduced by the authors to induce sparsity among model
parameters during the Bayesian inversion of nonlinear-in-parameter models. NSBL
relies on optimally selecting the hyperparameters of sparsity-inducing Gaussian
priors. It is inherently an approximate method since the uncertainty in the
hyperparameter posterior is disregarded as we instead seek the maximum a
posteriori (MAP) estimate of the hyperparameters (type-II MAP estimate). This
paper aims to investigate the hierarchical structure that forms the basis of
NSBL and validate its accuracy through a comparison with a one-level
hierarchical Bayesian inference as a benchmark in the context of three
numerical experiments: (i) a benchmark linear regression example with Gaussian
prior and Gaussian likelihood, (ii) the same regression problem with a highly
non-Gaussian prior, and (iii) an example of a dynamical system with a
non-Gaussian prior and a highly non-Gaussian likelihood function, to explore
the performance of the algorithm in these new settings. Through these numerical
examples, it can be shown that NSBL is well-suited for physics-based models as
it can be readily applied to models with non-Gaussian prior distributions and
non-Gaussian likelihood functions. Moreover, we illustrate the accuracy of the
NSBL algorithm as an approximation to the one-level hierarchical Bayesian
inference and its ability to reduce the computational cost while adequately
exploring the parameter posteriors.
