3 ms·

This is plain silly. If you don't have burn in, you bias your estimate. Sure, in the long run the bias becomes smaller and smaller, but, as the saying goes, in

This is plain silly. If you don't have burn in, you bias your estimate. Sure, in the long run the bias becomes smaller and smaller, but, as the saying goes, in the long run we are all dead. MCMC is generally slow, any way to speed it up a bit is good. Throwing away the burn-in part of the chain is just such a way to speed it up.

justindomke 6y ago
A helpful way to think about burn-in is that you probably don't care about bias as such but rather error which is a combination of bias and variance. Burn-in reduces bias at the cost of variance (since you have fewer points in your final estimates). My intuition is you want to be somewhere in the middle of the tradeoff. You want to tolerate some bias -- use burn-in to get partway but only partway to the stationary distribution. Here's a paper that advances this way of thinking: http://web.stanford.edu/~ngoodman/papers/LiederGriffithsGoodman2013NIPS.pdf http://web.stanford.edu/~ngoodman/papers/LiederGriffithsGood...
DiogenesKynikos 6y ago
The author's proposal to start at a high-probability-density (HPD) point reduces the bias that you get from not doing burn-in, but it still creates a bias. In order to get a reasonable answer from MCMC, you need to run for many correlation times. Burn-in should only last a few correlation times, so you don't lose that large a fraction of your chain by throwing it away. And the part of the chain you're using is biased, so you're probably not gaining anything by keeping it.
asdf_snar 6y ago
I think Geyer's point is that if you generate samples X_0, ..., X_b where b>0 is a burn-in time, and start collecting estimates at time t>b, then that's not somehow better than starting at X_b and not burning in (the two are equal in distribution).
- contravariant 6y ago
  The problem there is that (assuming a source of true noise) X_b is a random variable, in fact if b is sufficiently high its distribution approximates your target distribution pretty well. So if you're arguing "instead of burn-in you can also use an approximate sample of your target distribution" then that's technically correct. However generating an approximate sample is the exact problem that a MCMC process is trying to solve.
  - asdf_snar 6y ago
    Of course, you are right. I recall Geyer at some point writing some people thought there was a difference between having a burn-in period that happened to have X_b = x_b, and simply starting with X_0 = x_b. I can't find that on this page though.
    - contravariant 6y ago
      I see, you're right that those two are entirely equivalent (it's what makes the Markov chain 'Markov'). And using an approximate sample instead of burn-in isn't entirely without merit, though I think I've only seen that being used with things like Brownian motion where the trajectory is the thing of interest, rather than the samples themselves.
    - jmmcd 6y ago
      Yes, he writes "If you start at x and I start at x then your MCMC run is no better than mine." and surrounding text is making this point.
- derbOac 6y ago
  At that point, though, isn't it just semantics? If you have a chain, and throw out the beginning fraction, is that any more burnin than starting with X_b, which you probably had to find using some period of... burnin? This is really all about time series heterogeneity which is a topic unto itself. Eventually the chain reaches some stationary distribution but in the meantime there will be some shift in the distribution potentially. Seems like you could model the chain to do some test of where the distribution shifts, if at all. I imagine this is in the literature. Giving it a brief look I can't find anything but I haven't looked hard enough.

justindomke 6y ago

A helpful way to think about burn-in is that you probably don't care about bias as such but rather error which is a combination of bias and variance. Burn-in reduces bias at the cost of variance (since you have fewer points in your final estimates). My intuition is you want to be somewhere in the middle of the tradeoff. You want to tolerate some bias -- use burn-in to get partway but only partway to the stationary distribution. Here's a paper that advances this way of thinking: http://web.stanford.edu/~ngoodman/papers/LiederGriffithsGoodman2013NIPS.pdf http://web.stanford.edu/~ngoodman/papers/LiederGriffithsGood...

DiogenesKynikos 6y ago

The author's proposal to start at a high-probability-density (HPD) point reduces the bias that you get from not doing burn-in, but it still creates a bias. In order to get a reasonable answer from MCMC, you need to run for many correlation times. Burn-in should only last a few correlation times, so you don't lose that large a fraction of your chain by throwing it away. And the part of the chain you're using is biased, so you're probably not gaining anything by keeping it.

asdf_snar 6y ago

I think Geyer's point is that if you generate samples X_0, ..., X_b where b>0 is a burn-in time, and start collecting estimates at time t>b, then that's not somehow better than starting at X_b and not burning in (the two are equal in distribution).

contravariant 6y ago
The problem there is that (assuming a source of true noise) X_b is a random variable, in fact if b is sufficiently high its distribution approximates your target distribution pretty well. So if you're arguing "instead of burn-in you can also use an approximate sample of your target distribution" then that's technically correct. However generating an approximate sample is the exact problem that a MCMC process is trying to solve.
- asdf_snar 6y ago
  Of course, you are right. I recall Geyer at some point writing some people thought there was a difference between having a burn-in period that happened to have X_b = x_b, and simply starting with X_0 = x_b. I can't find that on this page though.
  - contravariant 6y ago
    I see, you're right that those two are entirely equivalent (it's what makes the Markov chain 'Markov'). And using an approximate sample instead of burn-in isn't entirely without merit, though I think I've only seen that being used with things like Brownian motion where the trajectory is the thing of interest, rather than the samples themselves.
  - jmmcd 6y ago
    Yes, he writes "If you start at x and I start at x then your MCMC run is no better than mine." and surrounding text is making this point.
derbOac 6y ago
At that point, though, isn't it just semantics? If you have a chain, and throw out the beginning fraction, is that any more burnin than starting with X_b, which you probably had to find using some period of... burnin? This is really all about time series heterogeneity which is a topic unto itself. Eventually the chain reaches some stationary distribution but in the meantime there will be some shift in the distribution potentially. Seems like you could model the chain to do some test of where the distribution shifts, if at all. I imagine this is in the literature. Giving it a brief look I can't find anything but I haven't looked hard enough.