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We describe an efficient sequential Monte Carlo method for sampling
multiway tables with given constraints, which can be used to approximate
exact conditional inference on contingency tables. An essential feature
of our new method is that it samples table entries sequentially according
to an appropriate proposal distribution. The sequential sampling approach
"divides and conquers" the difficult task of finding an appropriate
proposal distribution for a multiway table with complex constraints.
Computational commutative algebra is used to provide conditions that
guarantee that our method has certain good properties. We apply our method
to a range of examples from social and medical sciences to demonstrate its
efficiency in real problems.
