Bounds on inference

• Main Authors: Medard, Muriel (Contributor), Christiansen, Mark M. (Author), Duffy, Ken R. (Author), Tessaro, Stefano (Contributor), Calmon, Flavio du Pin (Contributor), Varia, Mayank H. (Contributor)
• Other Authors: Lincoln Laboratory (Contributor), Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory (Contributor), Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor), Massachusetts Institute of Technology. Research Laboratory of Electronics (Contributor)
• Format: Article
• Language: English
• Published: Institute of Electrical and Electronics Engineers (IEEE), 2014-09-29T16:26:42Z.
• Subjects: Article
• Online Access: Get fulltext

 Lower bounds for the average probability of error of estimating a hidden variable X given an observation of a correlated random variable Y, and Fano's inequality in particular, play a central role in information theory. In this paper, we present a lower bound for the average estimation error based on the marginal distribution of X and the principal inertias of the joint distribution matrix of X and Y. Furthermore, we discuss an information measure based on the sum of the largest principal inertias, called k-correlation, which generalizes maximal correlation. We show that k-correlation satisfies the Data Processing Inequality and is convex in the conditional distribution of Y given X. Finally, we investigate how to answer a fundamental question in inference and privacy: given an observation Y, can we estimate a function f(X) of the hidden random variable X with an average error below a certain threshold? We provide a general method for answering this question using an approach based on rate-distortion theory.