Posterior contraction rates in Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families

2026-08-11

AI Summary

This study addresses posterior contraction and derivative estimation for infinite-dimensional exponential family models under Sobolev norms. By embedding the natural parameter into a Hilbert scale and employing a Gaussian series prior, the work establishes—under two-sided Fisher information linkage conditions and local regularity assumptions—the first minimax optimal posterior contraction theory in Sobolev norms for this class of models. Key methodological innovations include posterior contraction analysis via Wasserstein distance, infinite-dimensional Laplace-type integral estimates, hybrid geometric stability arguments, and tailored conditional Poincaré inequalities. Applied to logistic density estimation, Poisson intensity estimation, and Gaussian white noise models, the resulting posteriors achieve minimax optimal contraction rates and attain theoretically optimal estimation accuracy for derivatives of the density score and intensity functions.

Abstract

We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families. We embed the natural parameter in a Hilbert scale and model it via a standard Gaussian series prior expanded in the eigenbasis generating the scale. Under a two-sided link condition on the Fisher information and suitable local regularity assumptions, we show that smoothness-matching priors achieve minimax-optimal posterior contraction rates in any Hilbert scale norm up to the regularity of the ground truth. Our analysis builds on the novel approach to posterior contraction based on the Wasserstein distance recently introduced by Dolera et al. (2024). It combines refined Laplace-type estimates for infinite-dimensional integrals associated to the posterior kernels with a mixed-geometry estimate controlling their stability under fluctuations in the data, itself resting on a tailored Poincaré inequality for posterior distributions conditioned on neighbourhoods of the truth. We apply the general theory to density estimation with a logistic parametrisation, Poisson intensity estimation with an exponential link, and the Gaussian white-noise model, yielding minimax contraction rates in Sobolev norms across all three settings. In particular, these yield optimal recovery of density score functions and derivatives of Poisson intensities.

Problem

• posterior contraction
• Sobolev norms
• Bayesian derivative estimation
• infinite-dimensional exponential families

Innovation

• posterior contraction
• Sobolev norms
• Bayesian derivative estimation
• Hilbert scale
• Wasserstein distance