Research Article
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Year 2022, Volume: 51 Issue: 1, 273 - 287, 14.02.2022
https://doi.org/10.15672/hujms.826558

Abstract

References

  • [1] H. Afşer, Statistical classification via robust hypothesis testing: non-asymptotic and simple bounds, IEEE Signal Process. Lett. 28, 2112-2116, 2021.
  • [2] J.I. Aizpurua, V.M. Catterson, B.G. Stewart and S.D.J. McArthur, Power transformer dissolved gas analysis through Bayesian networks and hypothesis testing, IEEE Trans. Dielectr. Electr. Insul. 25 (2), 494-506, 2018.
  • [3] U. Ali, M. Kieffer and P. Duhamel, Joint protocol-channel decoding for robust frame synchronization, IEEE Trans. Commun. 60 (8), 2326-2335, 2012.
  • [4] E. Biglieri and L. Gyorfi, Some remarks on robust binary hypothesis testing, IEEE International Symposium on Information Theory, Honolulu, Hawaii, 566-570, 2014.
  • [5] T.M. Cover and J.A. Thomas, Elements of Information Theory, John Wiley & Sons, 2012.
  • [6] L. Devroye, L. Gyorfi and G.A Lugosi, A note on robust hypothesis testing, IEEE Trans. Inf. Theory 48 (7), 2111-2014, 2002.
  • [7] S. Espinosa, J.F. Silva and P. Piantanida, Finite-length bounds on hypothesis testing subject to vanishing type I error restrictions, IEEE Signal Process. Lett. 28, 229-233, 2021.
  • [8] A.L. Gibbs and F.E. Su, On choosing and bounding the probability metrics, Int. Stat. Rev. 70 (3), 419-435, 2002.
  • [9] M. Gutman, Asymptotically optimal classification for multiple tests with empirically observed statistics , IEEE Trans. Inf. Theory 35 (2), 401-408, 1989.
  • [10] T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer-Verlag, 2009.
  • [11] P.J. Huber, A robust version of the probability ratio test, Ann. Math. Statist. 36 (6), 1753-1758, 1965.
  • [12] P.J. Huber, Robust Statistics, Wiley, 1981.
  • [13] M. Korki, H. Zayyani and J. Zhang, Bayesian hypothesis testing for block sparse signal recovery, IEEE Commun. Lett. 20 (3), 494-497, 2016.
  • [14] C.C. Leang and D.H. Johnson, On the asymptotics of M-hypothesis Bayesian detection, IEEE Trans. Inf. Theory 43 (1), 280-282, 1997.
  • [15] Y. Lee and Y. Sung, Generalized Chernoff information for mismatched Bayesian detection and its application to energy detection, IEEE Signal Process. Lett. 19 (11), 753-756, 2012.
  • [16] E.L. Lehmann and J.P. Romano, Testing Statistical Hypothesis, Springer-Verlag, 2005.
  • [17] C. Levy, Principles of Signal Detection and Parameter Estimation, Springer, 2008.
  • [18] A. Makris and C. Prieur, Bayesian multiple-hypothesis tracking of merging and splitting targets, IEEE Geosci. Remote Sens. Lett. 52 (12), 7684-7694, 2014.
  • [19] P. Moulin Asymptotically achievable error probabilities for multiple hypothesis testing, IEEE International Symposium on Information Theory, 1541-1545, 2016.
  • [20] I. Sason, Moderate deviations analysis of binary hypothesis testing, IEEE International Symposium on Information Theory, 821-825, 2012.
  • [21] I. Sason, Bounds on f-divergences and related distances, CCIT Report in Department of Electrical Engineering, Technion, Israel Institution of Technology, Haifa, Israel, 859, 2014.
  • [22] Y. Trachi, E. Elbouchikhi, V. Choqueuse, M.E.H. Benbouzid and T. Wang, A novel induction machine fault detector based on hypothesis testing, IEEE Trans. Ind Appl. 53 (3), 3039-3048, 2017.
  • [23] E. Tuncel, On error exponents in hypothesis testing, IEEE Trans. Inf. Theory 51 (8), 2945-2950, 2005.
  • [24] K.R. Varshney and L.R. Varshney, Quantization of prior probabilities for hypothesis testing, IEEE Trans. Signal Process. 56 (10), 4553-4562, 2008.
  • [25] M.B. Westover, Asymptotic geometry of multiple hypothesis Testing, IEEE Trans. Inf. Theory 54 (7), 3327-3329, 2008.
  • [26] U. Yıldırım and H. Afşer, A note on robust Bayesian multiple hypothesis testing, Eur J Lipid Sci Technol 24, 143-148, 2021.

Some remarks on Bayesian multiple hypothesis testing

Year 2022, Volume: 51 Issue: 1, 273 - 287, 14.02.2022
https://doi.org/10.15672/hujms.826558

Abstract

We consider Bayesian multiple hypothesis problem with independent and identically distributed observations. The classical, Sanov's theorem-based, analysis of the error probability allows one to characterize the best achievable error exponent. However, this analysis does not generalize to the case where the true distributions of the hypothesis are not exact or partially known via some nominal distributions. This problem has practical significance, because the nominal distributions may be quantized versions of the true distributions in a hardware implementation, or they may be estimates of the true distributions obtained from labeled training sequences as in statistical classification. In this paper, we develop a type-based analysis to investigate Bayesian multiple hypothesis testing problem. Our analysis allows one to explicitly calculate the error exponent of a given type and extends the classical analysis. As a generalization of the proposed method, we derive a robust test and obtain its error exponent for the case where the hypothesis distributions are not known but there exist nominal distribution that are close to true distributions in variational distance.

References

  • [1] H. Afşer, Statistical classification via robust hypothesis testing: non-asymptotic and simple bounds, IEEE Signal Process. Lett. 28, 2112-2116, 2021.
  • [2] J.I. Aizpurua, V.M. Catterson, B.G. Stewart and S.D.J. McArthur, Power transformer dissolved gas analysis through Bayesian networks and hypothesis testing, IEEE Trans. Dielectr. Electr. Insul. 25 (2), 494-506, 2018.
  • [3] U. Ali, M. Kieffer and P. Duhamel, Joint protocol-channel decoding for robust frame synchronization, IEEE Trans. Commun. 60 (8), 2326-2335, 2012.
  • [4] E. Biglieri and L. Gyorfi, Some remarks on robust binary hypothesis testing, IEEE International Symposium on Information Theory, Honolulu, Hawaii, 566-570, 2014.
  • [5] T.M. Cover and J.A. Thomas, Elements of Information Theory, John Wiley & Sons, 2012.
  • [6] L. Devroye, L. Gyorfi and G.A Lugosi, A note on robust hypothesis testing, IEEE Trans. Inf. Theory 48 (7), 2111-2014, 2002.
  • [7] S. Espinosa, J.F. Silva and P. Piantanida, Finite-length bounds on hypothesis testing subject to vanishing type I error restrictions, IEEE Signal Process. Lett. 28, 229-233, 2021.
  • [8] A.L. Gibbs and F.E. Su, On choosing and bounding the probability metrics, Int. Stat. Rev. 70 (3), 419-435, 2002.
  • [9] M. Gutman, Asymptotically optimal classification for multiple tests with empirically observed statistics , IEEE Trans. Inf. Theory 35 (2), 401-408, 1989.
  • [10] T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning: Data Mining, Inference, and Prediction, Springer-Verlag, 2009.
  • [11] P.J. Huber, A robust version of the probability ratio test, Ann. Math. Statist. 36 (6), 1753-1758, 1965.
  • [12] P.J. Huber, Robust Statistics, Wiley, 1981.
  • [13] M. Korki, H. Zayyani and J. Zhang, Bayesian hypothesis testing for block sparse signal recovery, IEEE Commun. Lett. 20 (3), 494-497, 2016.
  • [14] C.C. Leang and D.H. Johnson, On the asymptotics of M-hypothesis Bayesian detection, IEEE Trans. Inf. Theory 43 (1), 280-282, 1997.
  • [15] Y. Lee and Y. Sung, Generalized Chernoff information for mismatched Bayesian detection and its application to energy detection, IEEE Signal Process. Lett. 19 (11), 753-756, 2012.
  • [16] E.L. Lehmann and J.P. Romano, Testing Statistical Hypothesis, Springer-Verlag, 2005.
  • [17] C. Levy, Principles of Signal Detection and Parameter Estimation, Springer, 2008.
  • [18] A. Makris and C. Prieur, Bayesian multiple-hypothesis tracking of merging and splitting targets, IEEE Geosci. Remote Sens. Lett. 52 (12), 7684-7694, 2014.
  • [19] P. Moulin Asymptotically achievable error probabilities for multiple hypothesis testing, IEEE International Symposium on Information Theory, 1541-1545, 2016.
  • [20] I. Sason, Moderate deviations analysis of binary hypothesis testing, IEEE International Symposium on Information Theory, 821-825, 2012.
  • [21] I. Sason, Bounds on f-divergences and related distances, CCIT Report in Department of Electrical Engineering, Technion, Israel Institution of Technology, Haifa, Israel, 859, 2014.
  • [22] Y. Trachi, E. Elbouchikhi, V. Choqueuse, M.E.H. Benbouzid and T. Wang, A novel induction machine fault detector based on hypothesis testing, IEEE Trans. Ind Appl. 53 (3), 3039-3048, 2017.
  • [23] E. Tuncel, On error exponents in hypothesis testing, IEEE Trans. Inf. Theory 51 (8), 2945-2950, 2005.
  • [24] K.R. Varshney and L.R. Varshney, Quantization of prior probabilities for hypothesis testing, IEEE Trans. Signal Process. 56 (10), 4553-4562, 2008.
  • [25] M.B. Westover, Asymptotic geometry of multiple hypothesis Testing, IEEE Trans. Inf. Theory 54 (7), 3327-3329, 2008.
  • [26] U. Yıldırım and H. Afşer, A note on robust Bayesian multiple hypothesis testing, Eur J Lipid Sci Technol 24, 143-148, 2021.
There are 26 citations in total.

Details

Primary Language English
Subjects Statistics
Journal Section Statistics
Authors

Hüseyin Afşer 0000-0002-6302-4558

Publication Date February 14, 2022
Published in Issue Year 2022 Volume: 51 Issue: 1

Cite

APA Afşer, H. (2022). Some remarks on Bayesian multiple hypothesis testing. Hacettepe Journal of Mathematics and Statistics, 51(1), 273-287. https://doi.org/10.15672/hujms.826558
AMA Afşer H. Some remarks on Bayesian multiple hypothesis testing. Hacettepe Journal of Mathematics and Statistics. February 2022;51(1):273-287. doi:10.15672/hujms.826558
Chicago Afşer, Hüseyin. “Some Remarks on Bayesian Multiple Hypothesis Testing”. Hacettepe Journal of Mathematics and Statistics 51, no. 1 (February 2022): 273-87. https://doi.org/10.15672/hujms.826558.
EndNote Afşer H (February 1, 2022) Some remarks on Bayesian multiple hypothesis testing. Hacettepe Journal of Mathematics and Statistics 51 1 273–287.
IEEE H. Afşer, “Some remarks on Bayesian multiple hypothesis testing”, Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, pp. 273–287, 2022, doi: 10.15672/hujms.826558.
ISNAD Afşer, Hüseyin. “Some Remarks on Bayesian Multiple Hypothesis Testing”. Hacettepe Journal of Mathematics and Statistics 51/1 (February 2022), 273-287. https://doi.org/10.15672/hujms.826558.
JAMA Afşer H. Some remarks on Bayesian multiple hypothesis testing. Hacettepe Journal of Mathematics and Statistics. 2022;51:273–287.
MLA Afşer, Hüseyin. “Some Remarks on Bayesian Multiple Hypothesis Testing”. Hacettepe Journal of Mathematics and Statistics, vol. 51, no. 1, 2022, pp. 273-87, doi:10.15672/hujms.826558.
Vancouver Afşer H. Some remarks on Bayesian multiple hypothesis testing. Hacettepe Journal of Mathematics and Statistics. 2022;51(1):273-87.