Analytic solution of hierarchical variational Bayes in linear inverse problem

In singular models, the Bayes estimation, commonly, has the advantage of the generalization performance over the maximum likelihood estimation, however, its accurate approximation using Markov chain Monte Carlo methods requires huge computational costs. The variational Bayes (VB) approach, a tractable alternative, has recently shown good performance in the automatic relevance determination model (ARD), a kind of hierarchical Bayesian learning, in brain current estimation from magnetoencephalography (MEG) data, an ill-posed linear inverse problem. On the other hand, it has been proved that, in three-layer linear neural networks (LNNs), the VB approach is asymptotically equivalent to a positive-part James-Stein type shrinkage estimation. In this paper, noting the similarity between the ARD in a linear problem and an LNN, we analyze a simplified version of the VB approach in the ARD. We discuss its relation to the shrinkage estimation and how ill-posedness affects learning. We also propose the algorithm that requires simpler computation than, and will provide similar performance to, the VB approach.