The Randomness Recycler Approach to Perfect Sampling

1. The Randomness Recycler versus Markov chains At the heart of the Monte Carlo approach is the ability to sample from distributions that are in general very difficult to describe completely. For instance, the distribution might have an unknown normalizing constant which might require exponential time to compute. In these situations, in lieu of an exact approach, Markov chains are often employed to obtain approximately random samples. The primary drawback to Markov chain methods is that the mixing time of the chain is usually unknown, which makes it impossible to determine how close the output samples are to the target distribution. Here we present the randomness recycler (RR) protocol, which overcomes this difficulty for several problems of interest. In contrast to traditional Markov chain approaches, an RR-based algorithm creates samples that are drawn exactly from the desired distribution. Other perfect sampling methods such as coupling from the past use existing Markov chains, but RR does not use the traditional Markov chain at all. While not universal, RR does apply to a wide variety of problems. In restricted instances of these problems, it gives the first expected linear time algorithms for generating samples. Here we present RR-type algorithms for self-organizing lists, the Ising model, random independent sets, random colorings, and the random cluster model. In Markov chain approaches, small random changes are made in the observation until the entire observation has nearly the stationary distribution of the chain. The Metropolis [4] and heat bath algorithms utilize the idea of reversibility to design chains with a stationary distribution matching the desired distribution. Unfortunately, samples from the Markov chain approach are only approximately, not exactly, drawn from the stationary distribution of the chain. Moreover, they will not be close to the stationary distribution until a number of steps larger than the mixing time of the chain have been taken. Often the mixing time is unknown, and so the quality of the sample is suspect. Propp and Wilson have shown how to avoid these problems using techniques such as coupling from the past (CFTP) [5]. For some chains, CFTP provides a procedure that allows perfect samples to be drawn from the stationary distribution of the chain, without knowledge of the mixing time. However, CFTP and related approaches have drawbacks of their own. These algorithms are noninterruptible, which means that the user must commit to running such an