Sparsity and the Bayesian perspective

¹ Laboratoire AIM, UMR CEA-CNRS-Paris 7, Irfu, Service d’Astrophysique, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France
e-mail: jstarck@cea.fr
² Department of Statistics, Stanford University, Stanford CA, 94305, USA
³ GREYC CNRS-ENSICAEN-Université de Caen, 6 Bd du Maréchal Juin, 14050 Caen Cedex, France
⁴ Laboratoire d’Astrophysique, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, 1290 Versoix, Switzerland

Received: 7 February 2013
Accepted: 15 February 2013

Abstract

Sparsity has recently been introduced in cosmology for weak-lensing and cosmic microwave background (CMB) data analysis for different applications such as denoising, component separation, or inpainting (i.e., filling the missing data or the mask). Although it gives very nice numerical results, CMB sparse inpainting has been severely criticized by top researchers in cosmology using arguments derived from a Bayesian perspective. In an attempt to understand their point of view, we realize that interpreting a regularization penalty term as a prior in a Bayesian framework can lead to erroneous conclusions. This paper is by no means against the Bayesian approach, which has proven to be very useful for many applications, but warns against a Bayesian-only interpretation in data analysis, which can be misleading in some cases.