Bayesian Regression With Network Prior: Optimal Bayesian Filtering Perspective

The recently introduced intrinsically Bayesian robust filter (IBRF) provides fully optimal filtering relative to a prior distribution over an uncertainty class of joint random process models, whereas formerly the theory was limited to model-constrained Bayesian robust filters, for which optimization was limited to the filters that are optimal for models in the uncertainty class. This paper extends the IBRF theory to the situation where there are both a prior on the uncertainty class and sample data. The result is optimal Bayesian filtering (OBF), where optimality is relative to the posterior distribution derived from the prior and the data. The IBRF theories for effective characteristics and canonical expansions extend to the OBF setting. A salient focus of the present study is to demonstrate the advantages of Bayesian regression within the OBF setting over the classical Bayesian approach in the context of linear Gaussian models.