  This paper presents a sparse Bayesian learning (SBL) algorithm for linear
inverse problems with a high order total variation (HOTV) sparsity prior. For
the problem of sparse signal recovery, SBL often produces more accurate
estimates than maximum a posteriori estimates, including those that rely on l1
regularization. Moreover, rather than a single signal estimate, SBL yields a
full posterior density estimate which can be used for uncertainty
quantification. However, SBL is only immediately applicable to problems having
a direct sparsity prior, or to those that can be formed via synthesis. This
paper demonstrates how a problem with an HOTV sparsity prior can be formulated
via synthesis, and then develops a corresponding Bayesian learning method. This
expands the class of problems available to Bayesian learning to include, e.g.,
inverse problems dealing with the recovery of piecewise smooth functions or
signals from data. Numerical examples are provided to demonstrate how this new
technique is effectively employed.
