  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) as
well as f0 and g0.
  We first provide some identifiability result on this model and then adopt a
Bayesian point of view. In this view, we find some prior on f and g such that
the posterior distribution concentrates around the functions f0 and g0 when n
goes to infinity, we then obtain a contraction rate of order a power of
log(n)^(-1). We also obtain a lower bound on the model for the estimation of f0
and g0 in a frequentist paradigm which also decreases following a power of
log(n)^(-1).
