Optimal Bayesian feature selection (OBFS) is a multivariat e supervised screening method designed from the ground up for bioma rker discovery. In this work, we prove that Gaussian OBFS is strongly consisten t under mild conditions, and provide rates of convergence for key posteriors i n the framework. These results are of enormous importance, since they identify pre cisely what features are selected by OBFS asymptotically, characterize the relativ e rates of convergence for posteriors on different types of features, provide condi tions that guarantee convergence, justify the use of OBFS when its internal assum ptions are invalid, and set the stage for understanding the asymptotic behavior of other algorithms based on the OBFS framework.

Optimal Bayesian feature filtering (OBF) is a supervised screening method designed for biomarker discovery. In this article, we prove two major theoretical properties of OBF. First, optimal Bayesian feature selection under a general family of Bayesian models reduces to filtering if and only if the underlying Bayesian model assumes all features are mutually independent. Therefore, OBF is optimal if and only if one assumes all features are mutually independent, and OBF is the only filter method that is optimal under at least one model in the general Bayesian framework. Second, OBF under independent Gaussian models is consistent under very mild conditions, including cases where the data is non-Gaussian with correlated features. This result provides conditions where OBF is guaranteed to identify the correct feature set given enough data, and it justifies the use of OBF in non-design settings where its assumptions are invalid.