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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?

Greenparker
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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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Tea
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  • 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. – ekvall May 20 '16 at 14:25
  • @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. – Tea May 20 '16 at 14:57
  • 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. – ekvall May 20 '16 at 15:26
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    @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 ... – Tea May 20 '16 at 15:29