Abstract
We examine the robustness and privacy of Bayesian inference, under assumptions on the prior, and with no modifications to the Bayesian framework. First, we generalise the concept of differential privacy to arbitrary dataset distances, outcome spaces and distribution families. We then prove bounds on the robustness of the posterior, introduce a posterior sampling mechanism, show that it is differentially private and provide finite sample bounds for distinguishability-based privacy under a strong adversarial model. Finally, we give examples satisfying our assumptions.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Berger, J.O.: Statistical Decision Theory and Bayesian Analysis. Springer (1985)
Bickel, P.J., Doksum, K.A.: Mathematical Statistics: Basic Ideas and Selected Topics, vol. 1. Holden-Day Company (2001)
Bousquet, O., Elisseeff, A.: Stability and generalization. Journal of Machine Learning Research 2, 499–526 (2002)
Chatzikokolakis, K., Andrés, M.E., Bordenabe, N.E., Palamidessi, C.: Broadening the scope of differential privacy using metrics. In: De Cristofaro, E., Wright, M. (eds.) PETS 2013. LNCS, vol. 7981, pp. 82–102. Springer, Heidelberg (2013)
Chaudhuri, K., Hsu, D.: Convergence rates for differentially private statistical estimation. In: ICML (2012)
Chaudhuri, K., Monteleoni, C., Sarwate, A.D.: Differentially private empirical risk minimization. Journal of Machine Learning Research 12, 1069–1109 (2011)
DeGroot, M.H.: Optimal Statistical Decisions. John Wiley & Sons (1970)
Dimitrakakis, C., Nelson, B., Mitrokotsa, A., Rubinstein, B.: Robust and private Bayesian inference. Technical report, arXiv:1306.1066 (2014)
Duchi, J.C., Jordan, M.I., Wainwright, M.J.: Local privacy and statistical minimax rates. Technical report, arXiv:1302.3203 (2013)
Dwork, C.: Differential privacy. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4052, pp. 1–12. Springer, Heidelberg (2006)
Dwork, C., Lei, J.: Differential privacy and robust statistics. In: STOC, pp. 371–380 (2009)
Dwork, C., McSherry, F., Nissim, K., Smith, A.: Calibrating noise to sensitivity in private data analysis. In: Halevi, S., Rabin, T. (eds.) TCC 2006. LNCS, vol. 3876, pp. 265–284. Springer, Heidelberg (2006)
Dwork, C., Smith, A.: Differential privacy for statistics: What we know and what we want to learn. Journal of Privacy and Confidentiality 1(2), 135–154 (2009)
Fedotov, A.A., Harremoës, P., Topsoe, F.: Refinements of Pinsker’s inequality. IEEE Transactions on Information Theory 49(6), 1491–1498 (2003)
Grünwald, P.D., Dawid, A.P.: Game theory, maximum entropy, minimum discrepancy, and robust bayesian decision theory. The Annals of Statistics 32(4), 1367–1433 (2004)
Hall, R., Rinaldo, A., Wasserman, L.: Differential privacy for functions and functional data. Journal of Machine Learning Research 14, 703–727 (2013)
Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A.: Robust Statistics: The Approach Based on Influence Functions. John Wiley and Sons (1986)
Huber, P.J.: Robust Statistics. John Wiley and Sons (1981)
McSherry, F., Talwar, K.: Mechanism design via differential privacy. In: FOCS, pp. 94–103 (2007)
Mir, D.: Differentially-private learning and information theory. In: Proceedings of the 2012 Joint EDBT/ICDT Workshops, pp. 206–210. ACM (2012)
Norkin, V.: Stochastic Lipschitz functions. Cybernetics and Systems Analysis 22(2), 226–233 (1986)
Rubinstein, B.I.P., Bartlett, P.L., Huang, L., Taft, N.: Learning in a large function space: Privacy-preserving mechanisms for SVM learning. Journal of Privacy and Confidentiality 4(1) (2012)
Wasserman, L., Zhou, S.: A statistical framework for differential privacy. Journal of the American Statistical Association 105(489), 375–389 (2010)
Weissman, T., Ordentlich, E., Seroussi, G., Verdu, S., Weinberger, M.J.: Inequalities for the L1 deviation of the empirical distribution. Technical report, Hewlett-Packard Labs (2003)
Williams, O., McSherry, F.: Probabilistic inference and differential privacy. In: NIPS, pp. 2451–2459 (2010)
Xiao, Y., Xiong, L.: Bayesian inference under differential privacy. Technical report, arXiv:1203.0617 (2012)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Dimitrakakis, C., Nelson, B., Mitrokotsa, A., Rubinstein, B.I.P. (2014). Robust and Private Bayesian Inference. In: Auer, P., Clark, A., Zeugmann, T., Zilles, S. (eds) Algorithmic Learning Theory. ALT 2014. Lecture Notes in Computer Science(), vol 8776. Springer, Cham. https://doi.org/10.1007/978-3-319-11662-4_21
Download citation
DOI: https://doi.org/10.1007/978-3-319-11662-4_21
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11661-7
Online ISBN: 978-3-319-11662-4
eBook Packages: Computer ScienceComputer Science (R0)