Bayesian posterior consistency in the functional randomly shifted curves model

Dominique Bontemps (1) , Sébastien Gadat (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

In this paper, we consider the so-called Shape Invariant Model which stands for the estimation of a function f 0 submitted to a random translation of law g 0 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 . In this perspective, we adopt a Bayesian point of view and find prior on f and g such that the posterior distribution concentrates around P(f0,g0) at a polynomial rate when n goes to +∞. We obtain a logarithmic posterior contraction rate for the shape f0 and the distribution g 0 . We also derive logarithmic lower bounds for the estimation of f 0 and g 0 in a frequentist paradigm.

Mots clés: Minimax rate; Grenander's pattern theory; Shape Invariant Model; Mean pattern estimation; Non parametric Bayesian methods; Sobolev space; Minimax rate.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Dominique Bontemps, Sébastien Gadat. Bayesian posterior consistency in the functional randomly shifted curves model. 2013. ⟨hal-00768169v1⟩