Coupling from the past

Among Markov chain Monte Carlo (MCMC) algorithms, coupling from the past is a method for sampling from the stationary distribution of a Markov chain. Contrary to many MCMC algorithms, coupling from the past gives in principle a perfect sample from the stationary distribution. It was invented by James Propp and David Wilson in 1996.

The basic idea

Consider a finite state irreducible aperiodic Markov chain $M$ with state space $S$ and (unique) stationary distribution $\pi$ ($\pi$ is a probability vector). Suppose that we come up with a probability distribution $\mu$ on the set of maps $f:S\to S$ with the property that for every fixed $s\in S$, its image $f(s)$ is distributed according to the transition probability of $M$ from state $s$. An example of such a probability distribution is the one where $f(s)$ is independent from $f(s^{\prime })$ whenever $s\ne s^{\prime }$, but it is often worthwhile to consider other distributions. Now let $f_j$ for $j\in \mathbb{\mathbb{Z}}$ be independent samples from $\mu$.

Suppose that $x$ is chosen randomly according to $\pi$ and is independent from the sequence $f_j$. (We do not worry for now where this $x$ is coming from.) Then $f_{-1}(x)$ is also distributed according to $\pi$, because $\pi$ is $M$-stationary and our assumption on the law of $f$. Define

$F_j:=f_{-1}\circ f_{-2}\circ \cdots \circ f_{-j}.$

Then it follows by induction that $F_j(x)$ is also distributed according to $\pi$ for every $j\in \mathbb{\mathbb{N}}$. However, it may happen that for some $n\in \mathbb{\mathbb{N}}$ the image of the map $F_n$ is a single element of $S$. In other words, $F_n(x)=F_n(y)$ for each $y\in S$. Therefore, we do not need to have access to $x$ in order to compute $F_n(x)$. The algorithm then involves finding some $n\in \mathbb{\mathbb{N}}$ such that $F_n(S)$ is a singleton, and outputting the element of that singleton. The design of a good distribution $\mu$ for which the task of finding such an $n$ and computing $F_n$ is not too costly is not always obvious, but has been accomplished successfully in several important instances.

The monotone case

There is a special class of Markov chains in which there are particularly good choices for $\mu$ and a tool for determining if $|F_n(S)|=1$. (Here $|\cdot |$ denotes cardinality.) Suppose that $S$ is a partially ordered set with order $\le$, which has a unique minimal element $s_0$ and a unique maximal element $s_1$; that is, every $s\in S$ satisfies $s_0\le s\le s_1$. Also, suppose that $\mu$ may be chosen to be supported on the set of monotone maps $f:S\to S$. Then it is easy to see that $|F_n(S)|=1$ if and only if $F_n(s_0)=F_n(s_1)$, since $F_n$ is monotone. Thus, checking this becomes rather easy. The algorithm can proceed by choosing $n:=n_0$ for some constant $n_0$, sampling the maps $f_{-1},\dots ,f_{-n}$, and outputting $F_n(s_0)$ if $F_n(s_0)=F_n(s_1)$. If $F_n(s_0)\ne F_n(s_1)$ the algorithm proceeds by doubling $n$ and repeating as necessary until an output is obtained. (But the algorithm does not resample the maps $f_{-j}$ which were already sampled; it uses the previously sampled maps when needed.)

References

• Propp, James Gary; Wilson, David Bruce (1996). pp. 223–252. 
• Propp, James; Wilson, David (1998), "Coupling from the past: a user's guide", Microsurveys in discrete probability (Princeton, NJ, 1997), DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 41, Providence, R.I.: American Mathematical Society, pp. 181–192, doi:10.1090/dimacs/041/09, ISBN 9780821808276