Negative Examples for Sequential Importance
Sampling ofBinary Contingency Tables
Ivona Bezáková (RIT)Daniel Štefankovič (Rochester)
Alistair Sinclair (Berkeley)Eric Vigoda (Gatech)
The Voyage of the Beagle
Galápagos archipelago (1835)
Darwin’s Finches
© Robert H. Rothman
Darwin’s Finches
10
8
Darwin’s Finches
Darwin’s Finches
8
96
23781642
10
10
9 3 5 7 3 8 109 10 8
chance
OR
competitive pressures
?
Given: marginals (row sums, column sums)
Goal: • sample tables uniformly at random
• count tables
2
3
21
3
32
5
3 4 2
4
Binary Contingency Tables
Given: marginals (row sums, column sums)
Goal: • sample tables uniformly at random
• count tables
2
3
21
3
32
5
3 4 2
4
Binary Contingency Tables
Binary Contingency TablesGiven: marginals (row sums, column sums)
Goal: • sample tables uniformly at random
• count tables
2
3
21
3
32
5
3 4 2
4
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
2
3
21
3
32
5
3 4 2
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
2
3
22
3
12
5
4 3 3
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
2
3
22
3
12
5
4 3 3
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
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
2
3
3
5
4
4assign 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]
ri/(n-ri)
where product ranges over i: rows with assignment 1
Sequential Importance Sampling for BCT
1
2
2
5
4
3assign 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
ri/(n-ri)
Sequential Importance Sampling for BCT
1
2
2
5
4
3assign 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
ri/(n-ri)
Sequential Importance Sampling for BCT
0
1
2
4
4
3assign 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 3
ri/(n-ri)
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
3 3
ri/(n-ri)
0
1
2
4
3
Sequential Importance Sampling for BCT
0
1
1
3
4
2assign 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 3 2
ri/(n-ri)
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
3 3 2
ri/(n-ri)
0
1
1
3
2
Sequential Importance Sampling for BCT
0
1
1
2
4
1assign 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 3 22
ri/(n-ri)
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
3 3 22
ri/(n-ri)
0
1
1
2
1
Sequential Importance Sampling for BCT
0
1
0
1
4
1assign 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 3 22 2
ri/(n-ri)
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
3 3 22 2
ri/(n-ri)
0
1
0
1
1
Sequential Importance Sampling for BCT
0
1
0
0
4
0assign 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 3 22 2 1
ri/(n-ri)
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
3 3 22 2 1
ri/(n-ri)
0
1
0
0
0
Sequential Importance Sampling for BCT
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
3 3 22 2 1
ri/(n-ri)
0
0
0
0
0
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
3 3 22 2 1
ri/(n-ri)
2
3
3
5
4
A Counterexample for SIS
1 1 1 11 1 m
1
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).
A Counterexample for SIS
1 1 1 11 1
1
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
A Counterexample for SIS
1 1 1 11 1
1
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
A Counterexample for SIS
1 1 1 11 1
1
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
A Counterexample for SIS
1 1 1 11 1
1
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
A Counterexample for SIS
1 1 1 11 1
1
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
Expect: m ones
SIS: asymptotically fewer
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
SIS – Experimental Results
Bad example, m = 300, = 0.6, = 0.7
log
-scale
of
SIS
est
imate
number SIS steps
correct
SIS – Experimental Results
Regular marginals: m=50, marginals 5
SIS
est
imate
number SIS steps
correct
