A Hierarchical Likelihood Model for Non-linear Inverse Problems under Additive and Multiplicative Noise

Nicolas Goeman, Pierre-Antoine Thouvenin, Pierre Chainais

Ill-posed inverse problems are encountered in numerous applications, possibly characterized by a highly non-linear forward model, both additive and multiplicative sources of noise, and censored data. In the absence of ground truth, uncertainty quantification is crucial to assess estimation reliability. This motivates the use of a Bayesian model and stochastic inference methods such as Markov Chain Monte Carlo algorithms. Problems combining all these challenges often lead to a complex and potentially multimodal posterior distribution, difficult to handle in practice. Approximate approaches have been proposed in the literature by either neglecting a source of noise or by using a tractable approximation of the likelihood function. These approaches either lead to an inaccurate model, or may require a complex calibration of the approximate likelihood. This paper proposes to tackle such problems with a general hierarchical Bayesian model and an efficient MCMC algorithm. The proposed formulation bypasses the need for calibrating the hyperparameters of an approximate model and is more versatile. The proposed method is assessed on a challenging scenario encountered in astronomy using synthetic data, in a variety of noise and censoring configurations. Comparisons are conducted against two baselines and a state-of-the-art method applicable in this context. The proposed approach is general and yields state-of-the-art results in terms of point-wise estimates and computing costs, with superior predictive performance. Results in the supplementary material further complete this comprehensive and rigorous model study. This work can serve as a guide for practitioners to select the best likelihood model according to their specific application.