1. Negative Examples for Sequential Importance Sampling of Binary Contingency Tables Ivona Bezáková (RIT) Daniel Štefankovič (Rochester) Alistair Sinclair (Berkeley) Eric Vigoda (Gatech)

2. The Voyage of the Beagle Galápagos archipelago (1835) Darwin’s Finches

3. © Robert H. Rothman Darwin’s Finches

4. 10 8 Darwin’s Finches

5. 8 9 6 2 3 7 8 1 6 4 2 10 935738 9 8 chance OR competitive pressures ?

6. Given: marginals (row sums, column sums) Goal: sample tables uniformly at random count tables 2 3 21 3 32 5 342 4 Binary Contingency Tables

7. Given: marginals (row sums, column sums) Goal: sample tables uniformly at random count tables 2 3 21 3 32 5 342 4 Binary Contingency Tables

8. Given: marginals (row sums, column sums) Goal: sample tables uniformly at random count tables 2 3 21 3 32 5 342 4

9. Importance Sampling for counting problems x with positive probability  (x)>0 Probability distribution  on the points +   Random variable  (s) = 1/  (s) 0 if s in the set if s is  { Unbiased estimator E[  ] =   (x).1/  (x) = size of the set

10. 2 3 21 3 32 5 342 4 a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] Sequential Importance Sampling for BCT

11. 2 3 22 3 12 5 433 4 a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] Sequential Importance Sampling for BCT

12. 2 3 22 3 12 5 433 4 a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] Sequential Importance Sampling for BCT

13. 2 3 3 5 4 4 a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] Sequential Importance Sampling for BCT

14. 2 3 3 5 4 4 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05]  r i /(n-r i ) where product ranges over i: rows with assignment 1 Sequential Importance Sampling for BCT

15. 1 2 2 5 4 3 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 3  r i /(n-r i ) Sequential Importance Sampling for BCT

16. 1 2 2 5 4 3 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 3  r i /(n-r i ) Sequential Importance Sampling for BCT

17. 0 1 2 4 4 3 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 33  r i /(n-r i ) Sequential Importance Sampling for BCT

18. 4 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 33  r i /(n-r i ) 0 1 2 4 3 Sequential Importance Sampling for BCT

19. 0 1 1 3 4 2 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 332  r i /(n-r i ) Sequential Importance Sampling for BCT

20. 4 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 332  r i /(n-r i ) 0 1 1 3 2 Sequential Importance Sampling for BCT

21. 0 1 1 2 4 1 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 3322  r i /(n-r i ) Sequential Importance Sampling for BCT

22. 4 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 3322  r i /(n-r i ) 0 1 1 2 1 Sequential Importance Sampling for BCT

23. 0 1 0 1 4 1 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 33222  r i /(n-r i ) Sequential Importance Sampling for BCT

24. 4 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 33222  r i /(n-r i ) 0 1 0 1 1 Sequential Importance Sampling for BCT

25. 0 1 0 0 4 0 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 332221  r i /(n-r i ) Sequential Importance Sampling for BCT

26. 4 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 332221  r i /(n-r i ) 0 1 0 0 0 Sequential Importance Sampling for BCT

27. 4 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 332221  r i /(n-r i ) 0 0 0 0 0

28. Sequential Importance Sampling for BCT 4 assign the column with probability proportional to a specific  fill table column-by-column assign each column ignoring other column sums [Chen-Diaconis-Holmes-Liu ’05] where product ranges over i: rows with assignment 1 332221  r i /(n-r i ) 2 3 3 5 4

29. A Counterexample for SIS 111111 mm 1 mm 1 1 1 Thm [Bezáková-Sinclair-Štefankovič-Vigoda ‘06]: For any , SIS output after any subexponential number of trials is off by an exponential factor (with high probability).

30. A Counterexample for SIS 111111 1 mm 1 1 1 Thm [Bezáková-Sinclair-Štefankovič-Vigoda ‘06]: For any , SIS output after any subexponential number of trials is off by an exponential factor (with high probability). Intuition 1

31. A Counterexample for SIS 111111 1 mm 1 1 1 Thm [Bezáková-Sinclair-Štefankovič-Vigoda ‘06]: For any , SIS output after any subexponential number of trials is off by an exponential factor (with high probability). Intuition 1 Random table: - randomly choose  m ones

32. A Counterexample for SIS 111111 1 mm 1 1 1 Thm [Bezáková-Sinclair-Štefankovič-Vigoda ‘06]: For any , SIS output after any subexponential number of trials is off by an exponential factor (with high probability). Intuition 1 Random table: - randomly choose  m ones

33. A Counterexample for SIS 111111 1 mm 1 1 1 Thm [Bezáková-Sinclair-Štefankovič-Vigoda ‘06]: For any , SIS output after any subexponential number of trials is off by an exponential factor (with high probability). Intuition 1 Random table: - randomly choose  m ones

34. A Counterexample for SIS 111111 1 mm 1 1 1 Thm [Bezáková-Sinclair-Štefankovič-Vigoda ‘06]: For any , SIS output after any subexponential number of trials is off by an exponential factor (with high probability). Intuition 1 Random table: - randomly choose  m ones mm Expect:  m ones SIS: asymptotically fewer

35. A Counterexample for SIS Thm [Bezáková-Sinclair-Štefankovič-Vigoda ‘06]: For any , SIS output after any subexponential number of trials is off by an exponential factor (with high probability). Intuition Expect:  m ones SIS: asymptotically fewer all tables tables with ~  m ones tables seen by SIS whp

36. SIS – Experimental Results Bad example, m = 300,  = 0.6,  = 0.7 log-scale of SIS estimate number SIS steps correct

37. SIS – Experimental Results Regular marginals: m=50, marginals 5 SIS estimate number SIS steps correct