I often hear people often say they’re using a burn-in period in MCMC to run a Markov chain until it converges. But Markov chains don’t converge, at least not the Markov chains that are useful in MCMC. These Markov chains wander around forever exploring the domain they’re sampling from. Any point that makes a “bad” starting point for MCMC is a point you might reach by burn-in.
Not only that, Markov chains can’t remember how they got where they are. That’s their defining property. So if your burn-in period ends at a point x, the chain will perform exactly as if you had simply started at x.
When someone says a Markov chain has converged, they mean that the chain has entered a high-probability region. I’ll explain in a moment why that’s desirable. But the belief/hope is that a burn-in period will put a Markov chain in a high-probability region. And it probably will, but there are a couple reasons why this isn’t necessarily the best thing to do.
- Burn-in may be ineffective. You could use use optimization to be certain that you’re starting in such a region. Burn-in offers no such assurance. See Burn-in and other MCMC folklore.
- Burn-in may be inefficient. Casting aside worries that burn-in may not do what you want, it can be an inefficient way to find a high-probability region. MCMC isn’t designed to optimize a density function but rather to sample from it.
Why use burn-in? MCMC practitioners often don’t know how to do optimization, and in any case the corresponding optimization problem may be difficult. Also, if you’ve got the MCMC code in hand, it’s convenient to use it to find a starting point as well as for sampling.
So why does it matter whether you start your Markov chain in a high-probability region? In the limit, it doesn’t matter. But since you’re averaging some function of some finite number of samples, your average will be a better approximation if you start at a typical point in the density you’re sampling. If you start at a low probability location, your average may be more biased.
Samples from Markov chains don’t converge, but averages of functions applied to these samples may converge. When someone says a Markov chain has converged, they mean they’re at a point where the average of a finite number of function applications will be a better approximation of the thing they want to compute than if they’d started at a low probability point.
It’s not just a matter of imprecise language when people say a Markov chain has converged. It sometimes betrays a misunderstanding of how Markov chains work.