  We consider a dictionary learning problem whose objective is to design a
dictionary such that the signals admits a sparse or an approximate sparse
representation over the learned dictionary. Such a problem finds a variety of
applications such as image denoising, feature extraction, etc. In this paper,
we propose a new hierarchical Bayesian model for dictionary learning, in which
a Gaussian-inverse Gamma hierarchical prior is used to promote the sparsity of
the representation. Suitable priors are also placed on the dictionary and the
noise variance such that they can be reasonably inferred from the data. Based
on the hierarchical model, a variational Bayesian method and a Gibbs sampling
method are developed for Bayesian inference. The proposed methods have the
advantage that they do not require the knowledge of the noise variance \emph{a
priori}. Numerical results show that the proposed methods are able to learn the
dictionary with an accuracy better than existing methods, particularly for the
case where there is a limited number of training signals.
