4666 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 54, NO. 10, OCTOBER 2008 Successive Reﬁnement for Hypothesis Testing and Lossless One-Helper Problem Chao Tian, Member, IEEE, and Jun Chen, Member, IEEE Abstract—We investigate two closely related successive reﬁne- ment (SR) coding problems: 1) In the hypothesis testing (HT) problem, bivariate hypothesis against , i.e., test against independence is considered. One remote sensor collects data stream and sends summary information, constrained by SR coding rates, to a decision center which observes data stream directly. 2) In the one-helper (OH) problem, and are encoded separately and the receiver seeks to reconstruct losslessly. Multiple levels of coding rates are allowed at the two sensors, and the transmissions are performed in an SR manner. We show that the SR-HT rate-error-exponent region and the SR-OH rate region can be reduced to essentially the same entropy characterization form. Single-letter solutions are thus provided in a uniﬁed fashion, and the connection between them is discussed. These problems are also related to the information bottleneck (IB) problem, and through this connection we provide a straightforward operational meaning for the IB method. Connection to the pattern recognition problem, the notion of successive reﬁnability, and two speciﬁc sources are also discussed. A strong converse for the SR-HT problem is proved by generalizing the image size characterization method, which shows the optimal type-two error exponents under constant type-one error constraints are independent of the exact values of those constants. Index Terms—Entropy characterization, error exponent, hypothesis testing, image size characterization, information bot- tleneck, one-helper problem, successive reﬁnement. I. INTRODUCTION I N conventional successive reﬁnement (SR) source coding, a source stream is encoded into more than one description in a progressive order such that the later descriptions can be used to reﬁne the early ones, resulting in progressive reconstructions of improving qualities. As such, it can be conveniently formu- lated as a rate-distortion problem. In addition to the fundamental problem of characterizing the rate-distortion region, also of in- terest is the condition under which such a progressive coding requirement does not cause any performance loss, compared to a single stage coding system. These questions were the focus Manuscript received May 5, 2007; revised June 10, 2008. Current version published September 17, 2008. The material in this paper was presented in part at the IEEE International Symposium on Information Theory, Nice, France, June 2006. C. Tian was with the School of Computer and Communication Science, Ecole Polytechnique Fédérale de Lausanne, Lausanne, CH1015, Switzerland. He is now with the AT&T Labs–Research, Florham Park, NJ 07932 USA (e-mail: tian@research.att.com). J. Chen is with the Department of Electrical and Computer Engineering, Mc- Master University, Hamilton, ON L8S 4K1, Canada (e-mail: junchen@ece.mc- master.ca). Communicated by U. Mitra, Associate Editor At Large. Digital Object Identiﬁer 10.1109/TIT.2008.928951 Fig. 1. Hypothesis testing with one remote sensor. of early works [1]–[3]. The rate-distortion problem with var- ious extensions has subsequently been thoroughly researched, among which are the notable works by Effros [4], [5] and by Tuncel and Rose [6]–[8]. The successive reﬁnement coding structure is clearly ap- pealing in multimedia delivery systems, since such a framework allows a single copy of the multimedia content on the server to satisfy the requirement by users with different communication capabilities. However, the importance of successive reﬁnement coding goes well beyond this single speciﬁc application, and in the present work we investigate several such cases which deviate from the traditional rate-distortion setting. In the re- mainder of this section, we review related previous work on the hypothesis testing (HT) problem and the one-helper (OH) problem; the successive reﬁnement version of these problems in consideration and our contribution are also outlined. Formal problem deﬁnitions are given in Section II. A. The Hypothesis Testing Problem The information theoretic formulation of the hypothesis testing problem under communication constraint ﬁrst appeared in the award-winning article by Ahlswede and Csiszár [9], and the problem can be described as follows (see also Fig. 1). Source stream is observed by a remote sensor who communicates to the receiver under certain rate constraint , and the receiver, which observes another dependent source stream , wishes to distinguish between the two hypotheses and . The problem is to characterize the exponent of the type-two error ( is true but the detector judges oth- erwise), when the type-one error ( is true but the detector judges otherwise) is less than a pre-speciﬁed probability . For the case that , i.e., testing against inde- pendence, a single letter characterization of the error exponent was given in [9] for an arbitrary . This is the equiv- alence of the “strong converse” result encountered in Shannon theory as pointed out by Ahlswede and Csiszár, in comparison to the “weak converse” for which only the case is con- sidered. For a general alternative hypothesis , single letter lower and upper bounds were provided, yet a complete charac- terization was not found. Many subsequent works extended or strengthened the results in [9], for example, when both sensors are remote, or when the type-one error is constrained to satisfy 0018-9448/$25.00 © 2008 IEEE Authorized licensed use limited to: Chao Tian. Downloaded on February 19, 2009 at 10:02 from IEEE Xplore. Restrictions apply.