Survey Methodology
Bayesian differential privacy for small counties and individual commodities

by Balgobin Nandram, Habtamu Benecha and Linda J. Young^{Note 1}

Abstract

We construct a hybrid Bayesian method, which includes a differentially private mechanism, to mask Census county totals for a U.S. state on acreage of a commodity. We use surrogates for data collected at the farm level from a past U.S. Census of Agriculture to illustrate our procedure. We use two Bayesian small area models (parametric and mixture) to accommodate the smaller counties with fewer farms and some counties with large acres. In these models, the Laplace distribution provides a differentially private mechanism. In pre-processing, we also incorporate the Census weights to form the observed total acreage, a scaling factor to the Laplace mechanism for each county, a square-root transformation of the observed total acreage to avoid negative masked estimates especially for small counties, and the p-percent rule and the 3+ rule to partition the counties into suppressed counties, non-sensitive counties and sensitive counties. Because of difficulties in specifying and tuning the privacy budget (an unknown parameter), to balance security and utility, we specify a prior for the privacy budget, where the values are not specified, and the Gibbs sampler is used to fit the hierarchical Bayesian models. In post-processing, we use Bayesian predictive inference to obtain masked county acreages, and this includes a benchmarking so that the masked state total matches the observed state total. As a measure of reliability of the Bayesian procedure, we use the posterior coefficients of variation for the masked posterior means of the counties. As a measure of utility, we use the absolute relative errors for the individual counties, together with other global measures. For the sensitive counties, there are some differences between the two small area models but both are much better than an individual area model; the mixture model being the best compromise for security and utility.

Key Words: Bayesian predictive inference; Benchmarking; Laplace distribution; Mixture model with stick-breaking weights; p-percent rule; Utility; Post-processing.

Table of contents

• Section 1. Introduction
• Section 2. Background information
• Section 3. Bayesian models
• Section 4. Comparisons of the two models
• Section 5. Concluding remarks
• Disclaimer and Acknowledgements
• Appendix
• References

How to cite

Nandram, B., Benecha, H. and Young, L.J. (2026). Bayesian differential privacy for small counties and individual commodities. Survey Methodology, 52(1), 61-94. Available at: http://www.statcan.gc.ca/pub/12-001-x/2026001/article/00011-eng.pdf.

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