Adaptive posterior convergence rates in non-linear latent variable models

TL;DR

This paper investigates the theoretical rates at which the posterior distribution concentrates around the true density in non-linear latent variable models, providing new insights into their statistical properties.

Contribution

It introduces a novel analysis of posterior contraction rates for non-linear latent variable models with continuous mixing measures, extending existing results beyond mixture models.

Findings

Achieves near-optimal posterior contraction rates under regularity conditions.

Develops approximation results for continuous mixing measures.

Provides a theoretical foundation for Bayesian non-linear latent variable models.

Abstract

Non-linear latent variable models have become increasingly popular in a variety of applications. However, there has been little study on theoretical properties of these models. In this article, we study rates of posterior contraction in univariate density estimation for a class of non-linear latent variable models where unobserved U(0,1) latent variables are related to the response variables via a random non-linear regression with an additive error. Our approach relies on characterizing the space of densities induced by the above model as kernel convolutions with a general class of continuous mixing measures. The literature on posterior rates of contraction in density estimation almost entirely focuses on finite or countably infinite mixture models. We develop approximation results for our class of continuous mixing measures. Using an appropriate Gaussian process prior on the unknown regression function, we obtain the optimal frequentist rate up to a logarithmic factor under standard regularity conditions on the true density.