Bayesian posterior consistency in the functional randomly shifted curves model

TL;DR

This paper investigates Bayesian methods for estimating a function and its random translation law in a white noise model, establishing posterior concentration rates and lower bounds for the shape invariant model.

Contribution

It introduces a Bayesian approach with priors that achieve polynomial posterior concentration rates for the unknown function and translation law.

Findings

Posterior concentrates at a polynomial rate for the joint law and the function.

Logarithmic contraction rates are achieved for the shape and law.

Lower bounds confirm the optimality of the rates.

Abstract

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 $\PP_{f^0,g^0}$ as well as $f^0$ and $g^0$. In this perspective, we adopt a Bayesian point of view and find prior on $f$ and $g$ such that the posterior distribution concentrates around $\PP_{f^0,g^0}$ at a polynomial rate when $n$ goes to $+\infty$. We obtain a logarithmic posterior contraction rate for the shape $f^0$ and the distribution $g^0$. We also derive logarithmic lower bounds for the estimation of $f^0$ and $g^0$ in a frequentist paradigm.