  It has been shown both experimentally and theoretically that sparse signal
recovery can be significantly improved given that part of the signal's support
is known \emph{a priori}. In practice, however, such prior knowledge is usually
inaccurate and contains errors. Using such knowledge may result in severe
performance degradation or even recovery failure. In this paper, we study the
problem of sparse signal recovery when partial but partly erroneous prior
knowledge of the signal's support is available. Based on the conventional
sparse Bayesian learning framework, we propose a modified two-layer
Gaussian-inverse Gamma hierarchical prior model and, moreover, an improved
three-layer hierarchical prior model. The modified two-layer model employs an
individual parameter $b_i$ for each sparsity-controlling hyperparameter
$\alpha_i$, and has the ability to place non-sparsity-encouraging priors to
those coefficients that are believed in the support set. The three-layer
hierarchical model is built on the modified two-layer prior model, with a prior
placed on the parameters $\{b_i\}$ in the third layer. Such a model enables to
automatically learn the true support from partly erroneous information through
learning the values of the parameters $\{b_i\}$. Variational Bayesian
algorithms are developed based on the proposed hierarchical prior models.
Numerical results are provided to illustrate the performance of the proposed
algorithms.
