Bayesian methods for the shape invariant model

In this paper, we consider the so-called Shape Invariant Model that is used to model a function f⁰ submitted to a random translation of law g⁰ in a white noise. This model is of interest when the law of the deformations is unknown. Our objective is to recover the law of the process ℙ_f⁰,g⁰ as well as f⁰ and g⁰. To do this, we adopt a Bayesian point of view and find priors on f and g so that the posterior distribution concentrates at a polynomial rate around ℙ_f⁰,g⁰ when n goes to +∞. We then derive results on the identifiability of the SIM, as well as results on the functional objects themselves. We intensively use Bayesian non-parametric tools coupled with mixture models, which may be of independent interest in model selection from a frequentist point of view.