Random generation of combinatorial structures from a uniform
Theoretical Computer Science
Computational Statistics & Data Analysis
Random Structures & Algorithms
Sampling Binary Contingency Tables
Computing in Science and Engineering
Adaptive simulated annealing: A near-optimal connection between sampling and counting
Journal of the ACM (JACM)
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We describe multistage Markov chain Monte Carlo (MSMCMC) procedures which, in addition to estimating the total number of contingency tables with given positive row and column sums, estimate the number, $$Q$$Q, and the proportion, $$P$$P, of those tables that satisfy an additional, possibly, nonlinear constraint. Three Options, A, B, and C, are studied. Options A and B exploit locally optimal statistical properties whereas judicious assignment of a particular parameter of Option C allows estimation with approximately minimal standard error. Ten examples of varying dimensions and total entries illustrate and compare the procedures, where $$Q$$Q and $$P$$P denote the number and proportion of chi-squared statistics less than a given value. For both small and large dimensional tables, the comparisons favor Options A and B for moderate $$P$$P and Option C for small $$P$$P. Additional comparison with sequential importance sampling estimates favors the latter for small dimensional tables and moderate $$P$$P but favors Option C for large dimensional tables for both small and moderate $$P$$P. The proposed options extend an earlier MSMCMC technique for estimating total count and, in principle, can be further extended to incorporate additional constraints.