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In this paper, we consider the so-called Shape Invariant Model which standsfor the estimation of a function f0 submitted to a random translation of law g0in a white noise model. We are interested in such a model when the law of thedeformations is unknown. We aim to recover the law of the process P(f0,g0) aswell as f0 and g0.  We first provide some identifiability result on this model and then adopt aBayesian point of view. In this view, we find some prior on f and g such thatthe posterior distribution concentrates around the functions f0 and g0 when ngoes to infinity, we then obtain a contraction rate of order a power oflog(n)^(-1). We also obtain a lower bound on the model for the estimation of f0and g0 in a frequentist paradigm which also decreases following a power oflog(n)^(-1).

Bayesian methods in the Shape Invariant Model (II): Identifiability and posterior contraction rates on functional spaces