Bayesian Bandwidth Selection for Multivariate KDE via Expectation Propagation

Implements the approximate Bayesian method for bandwidth selection in multivariate kernel density estimation (KDE) proposed in Filippone & Sanguinetti (2011) . The method uses the Expectation Propagation (EP) algorithm to approximate the posterior distribution of the inverse bandwidth (precision matrix) under a leave-one-out cross-validated likelihood. Three covariance structures are supported: isotropic (scalar precision), diagonal, and full precision matrix. Online Bayesian updating is supported for the isotropic case. The approximate posterior can be used for bandwidth selection, model comparison (via the model evidence / Bayes factor), and online learning.