Asymptotic Estimation Performance of Linear Regression Model with Sparse Bayesian Learning as Both Samples and Signals Approach Infinity

This paper considers sparse Bayesian learning (SBL) in the linear regression model used in signal processing. An estimation performance of the method is analyzed in the asymptotic case where the sample size and the number of regression coefficients both increase. This subject has two cases: one is the dependence of performance on the hyperparameters in prior distribution and the other is how the performance depends on the sample size and the number of nonzero components in regression coefficients. To address this subject, we employ methods of statistical mechanics in physics. An equivalent of thermodynamic potential is calculated, following some similarities between Bayes inference and statistical mechanics. It is known in physics that the potential provides physical quantities. In a similar way, the equivalent of potential provides us with the asymptotic evaluation of estimation performance instead of physical quantities. The estimation performance in the two cases appears to have sharply divided regions in which the estimation succeeds and fails. These results are compared with related studies.