Sample continuation in bayesian hierarchical model via variational inference

Music City Center 

Posterior distributions in ill-posed Bayesian inverse problems are often analytically intractable and highly sensitive to prior assumptions. We study how a sample representation of the posterior evolves as prior parameters change, enabling sensitivity analysis for small perturbations and solution continuation for larger shifts. Our focus is on a class of non-conjugate hierarchical models that promote sparsity in linear inverse problems. These models, parameterized by a small set of shape parameters, encompass most classical sparsity-promoting priors. As parameters change, the posterior transitions from a tractable unimodal to an intractable multimodal distribution. To track these changes, we use Stein Variational Gradient Descent augmented with Birth-Death sampling, allowing efficient mass exchange between modes while optimizing kernel bandwidth. Our approach effectively samples multimodal posteriors and provides robust sensitivity analysis, as demonstrated in experimental results.

Bayesian Hierarchical model

Variational Inference

Distribution Evolution 

Section on Bayesian Statistical Science