独立伯努利变量和条件下的简单马尔可夫链
A simple Markov chain for independent Bernoulli variables conditioned on their sum
摘要：
我们考虑一个向量的$N$独立的二进制变量，每个成功的概率不同。以其和为条件的向量分布称为条件伯努利分布。假设$N$趋于无穷大，并且总和与$N$成比例，则精确采样的成本为$N^2$，而使用"交换"的简单马尔可夫链蒙特卡罗算法每次迭代的成本是恒定的。我们提供的条件下，这马尔可夫链收敛的顺序$N \log N$迭代。我们的证明依赖于耦合和辅助马尔可夫链上定义的空间划分成有利和不利的对。
英文摘要：
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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