Timeline for In exactly what sense do MCMC draws approximate the target?
Current License: CC BY-SA 4.0
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| Aug 8, 2018 at 21:48 | comment | added | user1237300 | I see. I will make a new question. Thanks Greenparker. Btw, any accessible reference to what you discussed in your answer? | |
| Aug 8, 2018 at 21:14 | comment | added | Greenparker | @user1237300 This sounds like a separate detailed question. If you post it as such, the community will be able to discuss it better. Also, clarify what you mean by "correlation length" | |
| Aug 8, 2018 at 21:10 | comment | added | user1237300 | Thanks for the clarification, Greenparker. I am curious to hear your thoughts on the following: It is advocated in Metropolis MCMC that the step size be chosen correspondingly to minimize the correlation length of the samples. It appears that this practice is to address objective 1 in the post above to make it comparable to standard Monte Carlo. But if your goal is the 2nd objective above, which is the histogram, what benefits would you have from minimizing the correlation length? is it for better exploration of the state space (since high correlation length means high accept rate?) | |
| Aug 8, 2018 at 16:06 | comment | added | Greenparker | @bamts I am not familiar with the literature on Glivenko-Cantelli, but a quick google search led me to this counter-example. So it would seem like ergodic processes don't share the uniform convergence. I am familiar with other results like the strong invariance principle etc holding under some stronger conditions on rate of convergence and moments of functions. | |
| Aug 8, 2018 at 16:04 | comment | added | jcz | Thanks for the answer! So, a follow-up concerning the ECDF $\hat{F}(\theta)$. As a consequence of (1), you get that $\hat{F}(\theta)$ is a valid pointwise approximation to $F(\theta)$. But in the iid case you have stronger results like the Glivenko-Cantelli theorem, which gives you uniform convergence. Does that still hold for MCMC? (Really what I'm thinking about is: "Convergence rates aside, do all of the statistical properties of iid draws carry over to MCMC draws?") | |
| Aug 8, 2018 at 13:45 | vote | accept | jcz | ||
| Aug 8, 2018 at 10:35 | history | answered | Greenparker | CC BY-SA 4.0 |