[Paper Review] Posterior convergence rates in non-linear latent variable models

Debdeep Pati, Anirban Bhattacharya+1 more|arXiv (Cornell University)|Sep 23, 2011

Bayesian Methods and Mixture Models16 references3 citations

TL;DR

This paper establishes posterior contraction rates for non-linear latent variable models in univariate density estimation, where latent uniform variables are transformed via a random non-linear function with Gaussian error. Using a Gaussian process prior on the regression function, the authors achieve the optimal frequentist rate up to a logarithmic factor under standard smoothness conditions, extending posterior contraction theory beyond discrete mixtures to continuous mixing measures.

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.

Motivation & Objective

• To study theoretical properties of non-linear latent variable models, particularly posterior contraction rates in density estimation.
• To extend existing posterior contraction theory—previously limited to finite or countably infinite mixtures—toward continuous mixing measures.
• To establish optimal posterior contraction rates for a class of non-linear latent variable models with unobserved U(0,1) latent variables and additive Gaussian noise.
• To develop approximation results for continuous mixing measures to support the theoretical analysis.
• To demonstrate that a Gaussian process prior on the non-linear regression function yields optimal frequentist rates up to a logarithmic factor.

Proposed method

• Model response variables as non-linear functions of latent U(0,1) variables with additive Gaussian error, forming a kernel convolution with a continuous mixing measure.
• Characterize the induced density space as a convolution of a kernel with a continuous mixing measure, enabling analysis beyond discrete mixtures.
• Use a Gaussian process prior on the unknown regression function to induce a prior on the mixing measure and ensure smoothness.
• Establish approximation results for the class of continuous mixing measures to control the distance between true and estimated densities.
• Apply techniques from posterior contraction theory, including entropy and testing arguments, under standard regularity conditions.
• Derive bounds on the posterior concentration using tail probabilities of Gaussian processes and entropy integral methods.

Experimental results

Research questions

• RQ1 What is the rate at which the posterior distribution contracts around the true density in non-linear latent variable models with continuous mixing measures?
• RQ2 Can posterior contraction rates for such models achieve the optimal frequentist rate up to a logarithmic factor?
• RQ3 How do Gaussian process priors on the regression function affect posterior concentration in this non-linear latent variable framework?
• RQ4 What approximation properties do continuous mixing measures possess in the context of density estimation?
• RQ5 How does the model avoid the clustering artifacts inherent in discrete mixture models while maintaining theoretical optimality?

Key findings

• The posterior contraction rate achieves the optimal frequentist rate up to a logarithmic factor under standard smoothness and regularity conditions.
• The model achieves optimal rates by using a Gaussian process prior on the non-linear regression function mapping latent U(0,1) variables to observed responses.
• The authors develop novel approximation results for continuous mixing measures, extending posterior contraction theory beyond discrete mixture models.
• The method avoids clustering artifacts common in finite or infinite discrete mixtures by using a continuous latent space.
• The posterior concentration bound is derived via entropy integral and tail probability techniques, showing that the posterior contracts at the desired rate.
• For sufficiently smooth true densities and appropriate choice of hyperparameters, the posterior concentrates in neighborhoods of the true density at the optimal rate, up to a logarithmic factor.