Negative Examples for Sequential Importance Sampling of Binary Contingency Tables Ivona Bezáková...

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

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





2

3

3

5

4

4

a specific





2

3

3

5

4

4assign the column with probability proportional to

a specific




ri/(n-ri)

where product ranges over i: rows with assignment 1


1

2

2

5

4


a specific





3

ri/(n-ri)


0

1

2

4

4


a specific





3 3

ri/(n-ri)


4

assign the column with probability proportional to

a specific





3 3

ri/(n-ri)

0

1

2

4

3


0

1

1

3

4


a specific





3 3 2

ri/(n-ri)


4


a specific





3 3 2

ri/(n-ri)

0

1

1

3

2


0

1

1

2

4


a specific





3 3 22

ri/(n-ri)


4


a specific





3 3 22

ri/(n-ri)

0

1

1

2

1


0

1

0

1

4


a specific





3 3 22 2

ri/(n-ri)


4


a specific





3 3 22 2

ri/(n-ri)

0

1

0

1

1


0

1

0

0

4


a specific





3 3 22 2 1

ri/(n-ri)


4


a specific





3 3 22 2 1

ri/(n-ri)

0

1

0

0

0



4


a specific





3 3 22 2 1

ri/(n-ri)

0

0

0

0

0


4


a specific





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).


1 1 1 11 1

1

m

1

1

1


For any , SIS output after any subexponential number of trials is off by an exponential factor (with high probability).Intuition

1


1 1 1 11 1

1

m

1

1

1



1

Random table:

- randomly choose m ones


1 1 1 11 1

1

m

1

1

1



1

Random table:

- randomly choose m ones

m

Expect: m ones

SIS: asymptotically fewer




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

Negative Examples for Sequential Importance Sampling of Binary Contingency Tables Ivona Bezáková...

Documents

Transcript of Negative Examples for Sequential Importance Sampling of Binary Contingency Tables Ivona Bezáková...