Electronic Journal of Statistics

Bayesian methods for the Shape Invariant Model

Dominique Bontemps and Sébastien Gadat

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In this paper, we consider the so-called Shape Invariant Model that is used to model a function $f^{0}$ submitted to a random translation of law $g^{0}$ in a white noise. This model is of interest when the law of the deformations is unknown. Our objective is to recover the law of the process $\mathbb{P}_{f^{0},g^{0}}$ as well as $f^{0}$ and $g^{0}$. To do this, we adopt a Bayesian point of view and find priors on $f$ and $g$ so that the posterior distribution concentrates at a polynomial rate around $\mathbb{P}_{f^{0},g^{0}}$ when $n$ goes to $+\infty$. We then derive results on the identifiability of the SIM, as well as results on the functional objects themselves. We intensively use Bayesian non-parametric tools coupled with mixture models, which may be of independent interest in model selection from a frequentist point of view.

Article information

Electron. J. Statist., Volume 8, Number 1 (2014), 1522-1568.

First available in Project Euclid: 8 September 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation 62F15: Bayesian inference
Secondary: 62G20: Asymptotic properties

Grenander’s pattern theory Shape Invariant Model Bayesian methods convergence rate of posterior distribution non-parametric estimation


Bontemps, Dominique; Gadat, Sébastien. Bayesian methods for the Shape Invariant Model. Electron. J. Statist. 8 (2014), no. 1, 1522--1568. doi:10.1214/14-EJS933. https://projecteuclid.org/euclid.ejs/1410181224

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