Robust Bayesian Inference for Measurement Error Misspecification: The Berkson and Classical Cases

Measurement error occurs when a covariate influencing a response variable is corrupted by noise. This can lead to misleading inference outcomes, particularly in problems where accurately estimating the relationship between covariates and response variables is crucial, such as causal effect estimation. Existing methods for dealing with measurement error often rely on strong assumptions such as knowledge of the error distribution or its variance and availability of replicated measurements of the covariates. We propose a Bayesian Nonparametric Learning framework that is robust to misspecification of these assumptions and does not require replicate measurements. This approach gives rise to a general framework that is suitable for both Classical and Berkson error models via the appropriate specification of the prior centering measure of a Dirichlet Process (DP). Moreover, it offers flexibility in the choice of loss function depending on the type of regression model. We provide bounds on the generalisation error based on the Maximum Mean Discrepancy (MMD) loss which allows for generalisation to non-Gaussian distributed errors and nonlinear covariate-response relationships. We showcase the effectiveness of the proposed framework versus prior art in real-world problems containing either Berkson or Classical measurement errors.