A simple Markov chain for independent Bernoulli variables conditioned on their sum
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We consider a vector of N independent binary variables, each with a different probability of success. The distribution of the vector conditional on its sum is known as the conditional Bernoulli distribution. Assuming that N goes to infinity and that the sum is proportional to N, exact sampling costs order N^2, while a simple Markov chain Monte Carlo algorithm using 'swaps' has constant cost per iteration. We provide conditions under which this Markov chain converges in order N log N iterations. Our proof relies on couplings and an auxiliary Markov chain defined on a partition of the space into favorable and unfavorable pairs.
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