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

TL;DR

This paper investigates the Bayesian estimation of functions and deformation laws in the Shape Invariant Model, establishing identifiability and deriving posterior contraction rates in a nonparametric setting.

Contribution

It provides new identifiability results and establishes posterior contraction rates for the Bayesian estimation of functions and deformation laws in the Shape Invariant Model.

Findings

Posterior concentrates around true functions at a log(n)^(-1) rate.

Identifiability of the model parameters is established.

Lower bounds for estimation accuracy are derived.

Abstract

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).