Sparsity and the Bayesian Perspective

TL;DR

This paper discusses the use of sparsity in cosmological data analysis, highlighting potential pitfalls of interpreting regularization as a Bayesian prior, and emphasizes the importance of careful interpretation beyond Bayesian frameworks.

Contribution

It clarifies misconceptions about Bayesian interpretation of sparsity regularization in cosmology and warns against over-reliance on Bayesian-only perspectives in data analysis.

Findings

Regularization penalties can be misinterpreted as Bayesian priors.

Bayesian interpretation may lead to erroneous conclusions in sparsity applications.

The paper advocates for cautious and nuanced interpretation beyond Bayesian frameworks.

Abstract

Sparsity has been recently introduced in cosmology for weak-lensing and 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, based on arguments derived from a Bayesian perspective. Trying 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 about a Bayesian-only interpretation in data analysis, which can be misleading in some cases.