Gaussian Processes with Errors in Variables: Theory and Computation
Authors: Shuang Zhou, Debdeep Pati, Tianying Wang, Yun Yang, Raymond J. Carroll; Volume 24, Issue 87, Pages 1-53, 2023.
Abstract
Covariate measurement error in nonparametric regression is a common issue in various fields such as nutritional epidemiology and geostatistics. While the frequentist literature has extensively studied this problem over the past two decades, Bayesian approaches for handling measurement error have only recently been explored and have shown surprising success. However, there is still a lack of proper theoretical justification for the asymptotic performance of the estimators. In this study, we propose a Gaussian process prior on the regression function and a Dirichlet process Gaussian mixture prior on the unknown distribution of the unobserved covariates. We demonstrate that the posterior distribution of the regression function and the unknown covariate density achieve optimal rates of contraction adaptively over a range of Holder classes, up to logarithmic terms. Additionally, we introduce a novel surrogate prior for approximating the Gaussian process prior, which enables efficient computation and preserves the covariance structure, facilitating easy prior elicitation. We provide empirical evidence of the performance of our approach through a wide range of simulation experiments and a real data example.
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