Differentially Private Bayesian Envelope Regression via Sufficient Statistic Perturbation

We propose a differentially private Bayesian framework for envelope regression, a technique that improves estimation efficiency by modelling the response as a function of a low-dimensional subspace of the predictors. Our method applies the analytic Gaussian mechanism to privatize sufficient statistics from the data, ensuring formal $(\epsilon ,\delta )$-differential privacy. We develop a tailored Gibbs sampling algorithm that performs valid Bayesian inference using only the noisy sufficient statistics. This approach leverages the envelope structure to isolate the variation in predictors that is relevant to the response, reducing estimation error compared to standard regression under the same privacy constraints. Through simulation studies, we demonstrate improved estimation accuracy and tighter credible intervals relative to a differentially private Bayesian linear regression baseline.