September 2011
Non-linear latent variable models have been increasingly popular in a variety of machine learning 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 $\mbox{U}(0,1)$ latent variables are related to the response variables via a random non-linear transformation 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 and using an appropriate Gaussian process prior on the unknown transformation, we obtain the optimal frequentist rate up to a logarithmic factor under standard regularity conditions on the true density.
Keywords: Asymptotics; Bayesian nonparametrics; Density estimation; Gaussian process; One factor model; Rate of convergence
The manuscript is available in PDF formats.