1. Stochastic Threshold Group Testing Chun Lam Chan, Sheng Cai, Mayank Bakshi, Sidharth Jaggi The Chinese University of Hong Kong Venkatesh Saligrama Boston University

2. q Classical Group testing For Pr(error)< ε, Lower bound of number of tests: What’s known [CCJS11] 2 q Chun Lam Chan; Pak Hou Che; Jaggi, S.; Saligrama, V.;, "Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms," 49th Annual Allerton Conference on Communication, Control, and Computing, pp.1832-1839, 28-30 Sept. 2011 [CCJS11] Adaptive vs. Non-adaptive

3. Classical vs. Stochastic Threshold 3 Stochastic Threshold Group Testing # defective items Prob. of positive outcome Classical Group Testing # defective items Prob. of positive outcome

4. 3 Stochastic Threshold Group Testing Classical Group Testing # defective items Prob. of positive outcome Classical vs. Stochastic Threshold Fair Coin # defective items Prob. of positive outcome

5. # defective items Prob. of positive outcome Linear vs. Bernoulli Gap Bernoulli Gap Linear Gap Biased Coin with more and more weight on positive 4 # defective items Prob. of positive outcome Noiseless Non-adaptive Bernoulli Gap

6. M. Jahangoshahi, S. Cai, M. Bakshi, S. Jaggi, “GROTESQUE: Noisy Group Testing (Quick and Efficient),” submitted to the IEEE Transactions on Information Theory, Mar. 2013 5 Our results: (1)Non-adaptive algorithm with Bernoulli gap model (2)Two-stage Adaptive algorithm (3)Non-adaptive algorithm with linear gap model Previous work: (NA) Non-adaptive algorithm (A) Adaptive algorithm * If not specified, the algorithm allows up to g misclassifications [CJBJ13] Faster implementation by [CJBJ13]

7. Transversal Design 6 Groups of the same size 1st family (partition) D. Balding, W. Bruno, D. Torney, and E. Knill, “A comparative survey of non-adaptive pooling designs,” in Genetic Mapping and DNA Sequencing, ser. The IMA Volumes in Mathematics and its Applications, T. Speed and M. Waterman, Eds. Springer New York, 1996, vol. 81, pp. 133–154. [BBTK96]

8. Transversal Design 6 2nd family (partition)

9. Transversal Design 6 3rd family (partition)

10. Transversal Design 6 Last family (partition)

11. positive Transversal Design Prob. of positive outcome Classical Group Testing # defective items 6 1st family (partition)

12. Transversal Design positive Prob. of positive outcome Classical Group Testing # defective items 6 2nd family (partition)

13. Transversal Design positive Prob. of positive outcome Classical Group Testing # defective items 6 3rd family (partition)

14. Transversal Design positive Prob. of positive outcome Classical Group Testing # defective items 6 Last family (partition) k

15. Transversal Design 6 100% positive tests Prob. of positive outcome Classical Group Testing # defective items

16. Transversal Design negative Prob. of positive outcome Classical Group Testing # defective items 6 1st family (partition)

17. Transversal Design negative Prob. of positive outcome Classical Group Testing # defective items 6 2nd family (partition)

18. Transversal Design 6 negative Prob. of positive outcome Classical Group Testing # defective items 3nd family (partition)

19. Transversal Design 6 positive Prob. of positive outcome Classical Group Testing # defective items Last family (partition)

20. Transversal Design 6 25% positive tests Prob. of positive outcome Classical Group Testing # defective items

21. Transversal Design 6 100% positive tests Prob. of positive outcome Classical Group Testing # defective items defective (higher ratio)

22. Transversal Design 6 25% positive tests Prob. of positive outcome Classical Group Testing # defective items non-defective (lower ratio) Statistical Difference!!

23. Transversal Design for “Stochastic” Group Testing 7 Fair Coin negative Prob. of positive outcome Classical Group Testing # defective items “Stochastic” Group Testing Fair Coin

24. Transversal Design for “Stochastic” Group Testing 8 100% positive tests Defective (higher ratio) 50% positive tests on expectation Prob. of positive outcome # defective items “Stochastic” Group Testing Fair Coin

25. Transversal Design for “Stochastic Group” Testing 8 25% positive tests non-defective (lower ratio) 12.5% positive tests on expectation Prob. of positive outcome # defective items “Stochastic” Group Testing Fair Coin Statistical Difference!! Statistical Difference # of families Union bound

26. Stochastic Threshold Group Testing negative 9 Stochastic Threshold Group Testing # defective items Prob. of positive outcome

27. Stochastic Threshold Group Testing Critical Reference group (R): Exactly l defectives Fair Coin negative 9 Stochastic Threshold Group Testing # defective items Prob. of positive outcome

28. Stochastic Threshold Group Testing “Stochastic” Group Testing # defective items Prob. of positive outcome 10 Critical Reference group (R): Exactly l defectives # defective items Prob. of positive outcome

29. Existence of Critical Reference Group 11 Size of reference group Prob. of picking l defectives ……

30. Identification of Critical Reference Group 12 1st family (partition)2nd family (partition) 3rd family (partition)Last family (partition)

31. 13 Identification of Critical Reference Group Exactly l defectives Less than l defectives More than l defectives

32. Identification of Critical Reference Group 14 1st family (partition)2nd family (partition) 3rd family (partition)Last family (partition) negative

33. Identification of Critical Reference Group 15 1st family (partition)2nd family (partition) 3rd family (partition)Last family (partition) positive

34. Identification of Critical Reference Group 16 1st family (partition)2nd family (partition) 3rd family (partition)Last family (partition) negative positive

35. 17 Identification of Critical Reference Group Exactly l defectives Critical! Less than l defectives More than l defectives 25% positive tests 75% positive tests 50% positive tests Statistical Difference!!

36. Threshold Group Testing with Linear Gap Smoother Gap, smaller statistical difference Many candidate critical reference groups # defective items Prob. of positive outcome 18

37. # defective items Prob. of positive outcome Other Gap As long as there exists a statistical difference 19 Threshold Group Testing with other Gap Models

38. THANK YOU 謝謝

39. Identification within Critical Reference Group is partitioned into and Reference groups are randomly picked from each complement set. 21

40. Threshold Group Testing Threshold Group Testing [Dam06] # defective items Prob. of positive outcome ?