Covariate measurement error in nonparametric regression is a common problem in nutritional epidemiology and geostatistics, and other fields. Over the last two decades, this problem has received substantial attention in the frequentist literature. Bayesian approaches for handling measurement error have only been explored recently and are surprisingly successful, although there still is a lack of a proper theoretical justification regarding the asymptotic performance of the estimators. By specifying a Gaussian process prior on the regression function and a Dirichlet process Gaussian mixture prior on the unknown distribution of the unobserved covariates, we show that the posterior distribution of the regression function and the unknown covariate density attain optimal rates of contraction adaptively over a range of Holder classes, up to logarithmic terms. We also develop a novel surrogate prior for approximating the Gaussian process prior that leads to efficient computation and preserves the covariance structure, thereby facilitating easy prior elicitation. We demonstrate the empirical performance of our approach and compare it with competitors in a wide range of simulation experiments and a real data example.

Transformation based methods have been an attractive approach in non-parametric inference for problems such as unconditioned and conditional density estimation due to their unique hierarchical structure that models the data as flexible transformation of a set of common latent variables. More recently, transformation based models have been used in variational inference (VI) to construct flexible implicit families of variational distributions. However, their use in both non-parametric inference and variational inference lacks theoretical justification. In the context of non-linear latent variable models (NL-LVM), we provide theoretical justification for the use of these models in non-parametric inference by showing that the support of the transformation induced prior in the space of densities is sufficiently large in the $L_1$ sense and show that for this class of priors the posterior concentrates at the optimal rate up to a logarithmic factor. Adopting the flexibility demonstrated in the non-parametric setting we use the NL-LVM to construct an implicit family of variational distributions, deemed as GP-IVI. We delineate sufficient conditions under which GP-IVI achieves optimal risk bounds and approximates the true posterior in the sense of the Kullback-Leibler divergence. To the best of our knowledge, this is the first work on providing theoretical guarantees for implicit variational inference.