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In some MCMC literature/source code, a Markov chain is often approximated with an AR(1) process. There is some theory to suggest that such an approximation is somewhat valid for a finite state space, but I am not aware of any literature for general state space.

See http://www.pnas.org/content/89/10/4432.full.pdf.

There is a lot of literature out there for approximating an AR process with a Markov chain, but I am interested in the opposite.

Most significantly where I have seen this approximation being used is the coda package in R, where it is used to estimate the spectral density at 0 for a process. Look at the help page for function spectrum0.ar here. This function is then used in the calculation of effective same size, amongst other things.

Seeing the popularity of this package, I am wondering how such an approximation is theoretically valid? Is there literature out there that justifies this approximation?

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AR(1) are Markov processes of a very particular kind. See:

Is AR(1) a Markov process?

However, not all Markov processes (also, not all MCMC) are of this form (see @NHR's answer again). Consequently, not all Markov chains can be reasonably well approximated with AR(1).

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  • $\begingroup$ You last conclusion doesn't follow from the statement before. Indeed, if all Markov processes where AR(1) to begin with, no approximations would be needed and the question wouldn't make sense. $\endgroup$ – ekvall May 20 '16 at 14:25
  • $\begingroup$ @Student001 A step in your logical deduction seems to be broken: (i) AR(1) $\neq$ All Markov Processes, in fact, AR(1) $\subset$ All Markov Processes, (ii) There are some Markov Processes that are from looking like an AR(1), (iii) Even if you have a Markov Process that is AR(1), finding the exact structure is not an easy task (you have to specify/estimate a parameter and the distribution of the "errors" $\epsilon$), that's why you would try to approximate it. $\endgroup$ – Tea May 20 '16 at 14:57
  • $\begingroup$ I don't contest any of that. Re iii) we seem to interpret the meaning of approximation here differently, that's fine. I'll try to make my main point more clear: There is no basis in your answer for the conclusion you draw after "consequently...". You have only argued that not all MCs have an exact AR representation. Whether or not they can all be approximated well (in some sense) remains to be shown. $\endgroup$ – ekvall May 20 '16 at 15:26
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    $\begingroup$ @Student001 I see. I think it is clear that you can construct a Markov process that looks nothing like an AR(1). But you are asking a little bit too much for this outlet, basically you want me to provide a distance between processes and to show that the minimum distance between the two kinds of processes is large enough to claim they cannot be used as an approximation ... $\endgroup$ – Tea May 20 '16 at 15:29

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