Bayesian Inference on Ratios Subject to Differentially Private Noise

Data privacy is a long-term issue for data sharing, especially in health-related data. Agencies need to find a balance between data utility and the privacy of the respondents. Differential privacy provides a solution by ensuring that the statistic of interest is basically the same, regardless of whether an individual is included or excluded. Due to this differentially private perturbation, analysts need to infer the true value from the released value. In this thesis, I propose Bayesian inference methods to infer the posterior distribution of ratios of two counts given the released values. I illustrate the Bayesian inference methods under several scenarios and with two commonly used differentially private mechanisms and prior distributions. The Bayesian inference method not only provides a point estimate but also provides posterior intervals. Simulation studies show that the Bayesian inference method can generate accurate inferences with close to nominal coverage rates, and have small values of bias and mean squared error.