Bayesian methods in the Shape Invariant Model (I): Posterior contraction rates on probability measures

In this paper, we consider the so-called Shape Invariant Model which stands for the estimation of a function f0 submitted to a random translation of law g0 in a white noise model. We are interested in such a model when the law of the deformations is unknown. We aim to recover the law of the process P(f0,g0). In this perspective, we adopt a Bayesian point of view and find prior on f and g such that the posterior distribution concentrates at a polynomial rate around P(f0,g0) when n goes to infinity. We intensively use some Bayesian non parametric tools coupled with mixture models and believe that some of our results obtained on this mixture framework may be also of interest for frequentist point of view.