Mixing Times 2 – Metropolis Chains

In our second reading group meeting for Mixing Times of Markov Chains, we reviewed chapters 3 and 4 of the Levin, Peres and Wilmer book. This post and the next contains a couple of brief thoughts about the ideas I found most interesting in each chapter.

Before reading chapter 3, the only thing I really knew about Monte Carlo methods was the slogan. If you want to sample from a probability distribution that you can’t describe explicitly, find a Markov chain which has that distribution as an equilibrium distribution, then run it for long enough starting from wherever you fancy. Then the convergence theorem for finite Markov chains means that the state of the chain after a long time approximates well the distribution you were originally looking for.

On the one previous occasion I had stopped and thought about this, I had two questions which I never really got round to answering. Firstly, what sort of distributions might you not be able to simulate directly? Secondly, and perhaps more fundamentally, how would you go about finding a Markov chain for which a given distribution is in equilibrium?

In the end, the second question is the one answered by this particular chapter. The method is called a Metropolis chain, and the basic idea is that you take ANY Markov chain with appropriate state space, then fiddle with the transition probabilities slightly. The starting chain is called a base chain. It is completely possible to adjust the following algorithm for a general base chain, but for simplicity, let’s assume it is possible to take an irreducible chain for which the transition matrix is symmetric. By thinking about the DBEs, this shows that the uniform distribution is the (unique) equilibrium distribution. Suppose the  transition matrix is given by $\Psi(x,y)$, to copy notation from the book. Then set:

$P(x,y)=\begin{cases}\Psi(x,y)\left[1\wedge \frac{\pi(y)}{\pi(x)}\right]&y\neq x\\ 1-\sum_{z\neq x} \Psi(x,z)\left [1\wedge \frac{\pi(z)}{\pi(x)}\right]& y=x.\end{cases}$

Note that this second case (y=x) is of essentially no importance. It just confirms that the rows of P add to 1. It is easy to check from the DBEs that $\pi$ is the equilibrium distribution of matrix P. One way to think of this algorithm is that we run the normal chain, but occasionally suppress transitions is they involve a move from a state which is likely (under $\pi$), to one which is less likely. This is done in proportion to the ratio, so it is unsurprising perhaps that the limit in distribution is $\pi$.

Conveniently, this algorithm also gives us some ideas for how to answer the first question. Note that at no point do we need to know $\pi(x)$ for some state x. We only need to use $\frac{\pi(x)}{\pi(y)}$ the ratios of probabilities. So this is perfect for distributions where there is a normalising constant which is computationally taxing to evaluate. For example, in the Ising model and similar statistical physics objects, probabilities are viewed more as weightings. There is a normalising constant, often called the partition function Z in this context, lying in the background, but especially the underlying geometry is quite exotic we definitely don’t want to have to worry about actually calculating Z. Thus we have a way to generate samples from such models. The other classic example is a random walk on a large, perhaps unknown graph. Then the equilibrium distribution at a vertex is inversely proportional to the degree of that vertex, but again you might not know about this information over the entire graph. It is reasonable to think of a situation where you might be able to take a random walk on a graph, say the connectivity graph of the internet, without knowing about all the edges at any one time. So, even though you potentially explore everywhere, you only need to know a small amount at any one time.

Of course, the drawback of both of these examples is that a lack of knowledge about the overall system means that it is hard in general to know how many steps the Metropolis chain must run before we can be sure that we are the equilibrium distribution it has been constructed to approach. So, while these chains are an excellent example to have in mind while thinking about mixing times, they are also a good motivation for the subject itself. General rules about speed of convergence to equilibrium are precisely what are required to make such implementation concrete.