Answer Bob's question on the paper: Sparse Signal Recovery with Temporally Correlated Source Vectors Using Sparse Bayesian Learning

Today I received an email from a reader, Bob, asking me why in high SNR cases or noiseless cases T-MSBL is better than T-SBL. What confused Bob is that in the paper T-MSBL is an approximation to T-SBL (using the approximation (20)). Both of them, in theory, are identical in the noiseless cases or the correlation-free cases. So, by intuition, T-MSBL should not be better than T-SBL, and in noiseless cases or correlation-free cases, T-MSBL has the same performance to T-SBL.

Thanks Bob for the good question.

First, I have to say, there is a slight difference between T-MSBL and T-SBL in noiseless cases or correlation-free cases. For T-SBL, the learning rule for B is given by Equation (13). For T-MSBL, the learning rule is given by Equation (28)-(29), which cannot be obtained from (13) using the approximation (20). As I stated in my paper, different B's can result in different performance. But I guess if T-MSBL adopts the rule (27), instead of (28)-(29), it should have the same performance to T-SBL in noiseless cases or correlation-free cases (because (27) is the simplified version of (13)).