The Annals of Statistics

Active sequential hypothesis testing

Mohammad Naghshvar and Tara Javidi

Full-text: Open access

Abstract

Consider a decision maker who is responsible to dynamically collect observations so as to enhance his information about an underlying phenomena of interest in a speedy manner while accounting for the penalty of wrong declaration. Due to the sequential nature of the problem, the decision maker relies on his current information state to adaptively select the most “informative” sensing action among the available ones.

In this paper, using results in dynamic programming, lower bounds for the optimal total cost are established. The lower bounds characterize the fundamental limits on the maximum achievable information acquisition rate and the optimal reliability. Moreover, upper bounds are obtained via an analysis of two heuristic policies for dynamic selection of actions. It is shown that the first proposed heuristic achieves asymptotic optimality, where the notion of asymptotic optimality, due to Chernoff, implies that the relative difference between the total cost achieved by the proposed policy and the optimal total cost approaches zero as the penalty of wrong declaration (hence the number of collected samples) increases. The second heuristic is shown to achieve asymptotic optimality only in a limited setting such as the problem of a noisy dynamic search. However, by considering the dependency on the number of hypotheses, under a technical condition, this second heuristic is shown to achieve a nonzero information acquisition rate, establishing a lower bound for the maximum achievable rate and error exponent. In the case of a noisy dynamic search with size-independent noise, the obtained nonzero rate and error exponent are shown to be maximum.

Article information

Source
Ann. Statist., Volume 41, Number 6 (2013), 2703-2738.

Dates
First available in Project Euclid: 17 December 2013

Permanent link to this document
https://projecteuclid.org/euclid.aos/1387313387

Digital Object Identifier
doi:10.1214/13-AOS1144

Mathematical Reviews number (MathSciNet)
MR3161445

Zentralblatt MATH identifier
1292.62037

Subjects
Primary: 62F03: Hypothesis testing
Secondary: 62B10: Information-theoretic topics [See also 94A17] 62B15: Theory of statistical experiments 62L05: Sequential design

Keywords
Active hypothesis testing sequential analysis optimal stopping dynamic programming feedback gain error exponent information acquisition rate

Citation

Naghshvar, Mohammad; Javidi, Tara. Active sequential hypothesis testing. Ann. Statist. 41 (2013), no. 6, 2703--2738. doi:10.1214/13-AOS1144. https://projecteuclid.org/euclid.aos/1387313387


Export citation

References

  • [1] Albert, A. E. (1961). The sequential design of experiments for infinitely many states of nature. Ann. Math. Statist. 32 774–799.
  • [2] Armitage, P. (1950). Sequential analysis with more than two alternative hypotheses, and its relation to discriminant function analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 12 137–144.
  • [3] Bartroff, J. (2007). Asymptotically optimal multistage tests of simple hypotheses. Ann. Statist. 35 2075–2105.
  • [4] Berlekamp, E. R. (1964). Block coding with noiseless feedback. Ph.D. thesis, MIT, Cambridge, MA.
  • [5] Berlin, P., Nakiboğlu, B., Rimoldi, B. and Telatar, E. (2009). A simple converse of Burnashev’s reliability function. IEEE Trans. Inform. Theory 55 3074–3080.
  • [6] Berry, S. M., Carlin, B. P., Lee, J. J. and Müller, P. (2011). Bayesian Adaptive Methods for Clinical Trials. Chapman & Hall/CRC Biostatistics Series 38. CRC Press, Boca Raton, FL.
  • [7] Bertsekas, D. P. and Shreve, S. E. (2007). Stochastic Optimal Control: The Discrete-Time Case. Athena Scientific, Belmont, CA.
  • [8] Bessler, S. A. (1960). Theory and applications of the sequential design of experiments, $K$-actions and infinitely many experiments: Part I—Theory. Technical Report no. 55, Dept. Statistics, Univ. Stanford, Stanford, CA.
  • [9] Blackwell, D. (1953). Equivalent comparisons of experiments. Ann. Math. Statist. 24 265–272.
  • [10] Blahut, R. E. (1974). Hypothesis testing and information theory. IEEE Trans. Inform. Theory 20 405–417.
  • [11] Blot, W. J. and Meeter, D. A. (1973). Sequential experimental design procedures. J. Amer. Statist. Assoc. 68 586–593.
  • [12] Burnashev, M. V. (1975). Data transmission over a discrete channel with feedback. Random transmission time. Problemy Peredachi Informatsii 12 10–30.
  • [13] Burnashev, M. V. (1980). Sequential discrimination of hypotheses with control of observations. Mathematics of the USSR–Izvestiya 15 419–440.
  • [14] Burnashev, M. V. and Zigangirov, K. S. (1974). A certain problem of interval estimation in observation control. Problemy Peredachi Informatsii 10 51–61.
  • [15] Castanon, D. A. (1995). Optimal search strategies in dynamic hypothesis testing. IEEE Trans. Syst., Man Cybern. 25 1130–1138.
  • [16] Chan, C. L., Che, P. H., Jaggi, S. and Saligrama, V. (2011). Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms. In 49th Annual Allerton Conference on Communication, Control, and Computing 1832–1839.
  • [17] Chernoff, H. (1959). Sequential design of experiments. Ann. Math. Statist. 30 755–770.
  • [18] Cover, T. M. and Thomas, J. A. (2006). Elements of Information Theory, 2nd ed. Wiley, Hoboken, NJ.
  • [19] Csiszár, I. and Shields, P. C. (2004). Information theory and statistics: A tutorial. Found. Trends Commun. Inf. Theory 1 417–528.
  • [20] DeGroot, M. H. (1962). Uncertainty, information, and sequential experiments. Ann. Math. Statist. 33 404–419.
  • [21] DeGroot, M. H. (1970). Optimal Statistical Decisions. McGraw-Hill, New York.
  • [22] Dragalin, V. P., Tartakovsky, A. G. and Veeravalli, V. V. (1999). Multihypothesis sequential probability ratio tests. I. Asymptotic optimality. IEEE Trans. Inform. Theory 45 2448–2461.
  • [23] Gallager, R. G. (1968). Information Theory and Reliable Communication. Wiley, New York.
  • [24] Goel, P. K. and DeGroot, M. H. (1979). Comparison of experiments and information measures. Ann. Statist. 7 1066–1077.
  • [25] Haroutunian, E. A., Haroutunian, M. E. and Harutyunyan, A. N. (2007). Reliability criteria in information theory and in statistical hypothesis testing. Found. Trends Commun. Inf. Theory 4 97–263.
  • [26] Hayashi, M. (2009). Discrimination of two channels by adaptive methods and its application to quantum system. IEEE Trans. Inform. Theory 55 3807–3820.
  • [27] Hero, A. O. and Cochran, D. (2011). Sensor management: Past, present, and future. IEEE Sens. J. 11 3064–3075.
  • [28] Hollinger, G. A., Mitra, U. and Sukhatme, G. S. (2011). Active classification: Theory and application to underwater inspection. In Proceedings of the 15th International Symposium on Robotics Research (ISRR), August 28–September 1, 2011. Flagstaff, AZ.
  • [29] Horstein, M. (1963). Sequential transmission using noiseless feedback. IEEE Trans. Inform. Theory 9 136–143.
  • [30] Kadane, J. B. (1971). Optimal whereabouts search. Oper. Res. 19 894–904.
  • [31] Keener, R. (1984). Second order efficiency in the sequential design of experiments. Ann. Statist. 12 510–532.
  • [32] Kiefer, J. and Sacks, J. (1963). Asymptotically optimum sequential inference and design. Ann. Math. Statist. 34 705–750.
  • [33] Kumar, P. R. and Varaiya, P. (1986). Stochastic Systems: Estimation, Identification, and Adaptive Control. Prentice-Hall, Upper Saddle River, NJ.
  • [34] Lalley, S. P. and Lorden, G. (1986). A control problem arising in the sequential design of experiments. Ann. Probab. 14 136–172.
  • [35] Le Cam, L. (1964). Sufficiency and approximate sufficiency. Ann. Math. Statist. 35 1419–1455.
  • [36] Lehmann, E. L. (1988). Comparing location experiments. Ann. Statist. 16 521–533.
  • [37] Lindley, D. V. (1956). On a measure of the information provided by an experiment. Ann. Math. Statist. 27 986–1005.
  • [38] Lorden, G. (1970). On excess over the boundary. Ann. Math. Statist. 41 520–527.
  • [39] Lorden, G. (1977). Nearly-optimal sequential tests for finitely many parameter values. Ann. Statist. 5 1–21.
  • [40] Lorden, G. (1983). Asymptotic efficiency of three-stage hypothesis tests. Ann. Statist. 11 129–140.
  • [41] McDiarmid, C. (1989). On the method of bounded differences. In Surveys in Combinatorics, 1989 (Norwich, 1989). London Mathematical Society Lecture Note Series 141 148–188. Cambridge Univ. Press, Cambridge.
  • [42] Nachlas, J. A., Loney, S. R. and Binney, B. A. (1990). Diagnostic-strategy selection for series systems. IEEE Trans. Reliab. 39 273–280.
  • [43] Naghshvar, M. and Javidi, T. (2010). Active $M$-ary sequential hypothesis testing. In Proceedings of the IEEE International Symposium on Information Theory (ISIT), 1318 June 2010 1623–1627, Austin, TX.
  • [44] Naghshvar, M. and Javidi, T. (2013). Supplement to “Active sequential hypothesis testing.” DOI:10.1214/13-AOS1144SUPP.
  • [45] Nakiboğlu, B. and Gallager, R. G. (2008). Error exponents for variable-length block codes with feedback and cost constraints. IEEE Trans. Inform. Theory 54 945–963.
  • [46] Nakiboǧlu, B. and Zheng, L. (2012). Errors-and-erasures decoding for block codes with feedback. IEEE Trans. Inform. Theory 58 24–49.
  • [47] Nitinawarat, S., Atia, G. and Veeravalli, V. V. (2013). Controlled sensing for multihypothesis testing. IEEE Trans. Automat. Control 58 2451–2464.
  • [48] Nowak, R. D. (2011). The geometry of generalized binary search. IEEE Trans. Inform. Theory 57 7893–7906.
  • [49] Polyanskiy, Y., Poor, H. V. and Verdú, S. (2011). Feedback in the non-asymptotic regime. IEEE Trans. Inform. Theory 57 4903–4925.
  • [50] Polyanskiy, Y. and Verdu, S. (2011). Hypothesis testing with feedback. In Information Theory and Applications Workshop (ITA), 611 February 2011, San Diego, CA.
  • [51] Posner, E. (1963). Optimal search procedures. IEEE Trans. Inform. Theory 9 157–160.
  • [52] Sejdinovic, D. and Johnson, O. (2010). Note on noisy group testing: Asymptotic bounds and belief propagation reconstruction. In Proceedings of the 48th Annual Allerton Conference on Communication, Control, and Computing, September 29–October 1, 2010 998–1003, Monticello, IL.
  • [53] Shannon, C. E. (1956). The zero error capacity of a noisy channel. Institute of Radio Engineers Transactions on Information Theory 2 8–19.
  • [54] Shenoy, P. and Yu, A. J. (2011). Rational decision-making in inhibitory control. Front. Human Neurosci. 5 48.
  • [55] Stone, L. D. (1975). Theory of Optimal Search. Academic Press, New York.
  • [56] Tognetti, K. P. (1968). An optimal strategy for a whereabouts search. Oper. Res. 16 209–211.
  • [57] Torgersen, E. (1991). Stochastic orders and comparison of experiments. In Stochastic Orders and Decision Under Risk (Hamburg, 1989). Institute of Mathematical Statistics Lecture Notes—Monograph Series 19 334–371. IMS, Hayward, CA.
  • [58] Wald, A. and Wolfowitz, J. (1948). Optimum character of the sequential probability ratio test. Ann. Math. Statist. 19 326–339.

Supplemental materials

  • Supplementary material: Technical proofs. For the interest of space, we only provided the proofs of the theorems in this paper. Proofs of the propositions, lemmas, corollaries and technical claims are provided in the supplemental article.