Locally Private Causal Inference

Local differential privacy (LDP) is a differential privacy (DP) paradigm in which individuals first apply a DP mechanism to their data (often by adding noise) before transmitting the result to a curator. In this article, we develop methodologies to infer causal effects from locally privatized data under the Rubin Causal Model framework. First, we present frequentist estimators under various privacy scenarios with their variance estimators and plug-in confidence intervals. We show that using a plug-in estimator results in inferior mean-squared error (MSE) compared to minimax lower bounds. In contrast, we show that using a customized privacy mechanism, we can match the lower bound, giving minimax optimal inference. We also develop a Bayesian nonparametric methodology along with a blocked Gibbs sampling algorithm, which can be applied to any of our proposed privacy mechanisms, and which performs especially well in terms of MSE for tight privacy budgets. Finally, we present simulation studies to evaluate the performance of our proposed frequentist and Bayesian methodologies for various privacy budgets, resulting in useful suggestions for performing causal inference for privatized data.