Negative Examples for Sequential Importance Sampling of Binary 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

chance OR competitive pressures ?

Given: marginals (row sums, column sums) Goal: sample tables uniformly at random count tables Binary Contingency Tables

Given: marginals (row sums, column sums) Goal: sample tables uniformly at random count tables Binary Contingency Tables

Given: marginals (row sums, column sums) Goal: sample tables uniformly at random count tables

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

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

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

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

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

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

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

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

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

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 ) Sequential Importance Sampling for BCT

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  r i /(n-r i ) 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  r i /(n-r i ) Sequential Importance Sampling for BCT

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  r i /(n-r i ) 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  r i /(n-r i ) Sequential Importance Sampling for BCT

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  r i /(n-r i ) 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  r i /(n-r i ) Sequential Importance Sampling for BCT

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  r i /(n-r i ) 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  r i /(n-r i ) 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  r i /(n-r i )

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  r i /(n-r i )

A Counterexample for SIS mm 1 mm 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 mm 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 mm 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 mm 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 mm 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 mm 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

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 estimate number SIS steps correct

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