A simple algorithm for estimating distribution parameters from n-dimensional randomized binary responses

Randomized response for privacy protection is attractive as provided disclosure control can be quantified by means such as differential privacy. However, recovering statistics involving multiple dependent binary attributes can be difficult, posing a barrier to the use of randomized response for privacy protection. In this work, we identify a family of randomizers for which we are able to present a simple and efficient algorithm for obtaining unbiased maximum likelihood estimates for k-way marginal distributions from the randomized data. We also provide theoretical bounds on the statistical efficiency of these estimates, allowing the assessment of sample sizes for these randomizers. The identified family consists of randomizers generated by an iterated Kronecker product of an invertible and bisymmetric 2×2 matrix. This family includes modes of Google’s Rappor randomizer, as well as applications of two well-known classical randomized response methods: Warner’s original method, and Simmons’ unrelated question method. We find that randomizers in this family can also be considered to be equivalent to each other with respect to the efficiency – differential privacy tradeoff. Importantly, the estimation algorithm is simple to implement, an aspect critical to technologies for privacy protection and security.