  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 mismeasured
covariates, does not require the preceding assumptions, and can incorporate
prior beliefs about the error distribution. 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
generalization error based on the Maximum Mean Discrepancy (MMD) loss which
allows for generalization 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.
