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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 targe
by 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 agoOf 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 agoI 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 agoYes, 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.