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

Posterior convergence rates in non-linear latent variable models