Maximum-a-posteriori estimation with unknown regularisation parameters

This paper presents two hierarchical Bayesian methods for performing maximum-a-posteriori inference when the value of the regularisation parameter is unknown. The methods are useful for models with homogenous regularisers (i.e., prior sufficient statistics), including all norms, composite norms and compositions of norms with linear operators. A key contribution of this paper is to show that for these models the normalisation factor of the prior has a closed-form analytic expression. This then enables the development of Bayesian inference techniques to either estimate regularisation parameters from the observed data or, alternatively, to remove them from the model by marginalisation followed by inference with the marginalised model. The effectiveness of the proposed methodologies is illustrated on applications to compressive sensing using an l1-wavelet analysis prior, where they outperform a state-of-the-art SURE-based technique, both in terms of estimation accuracy and computing time.