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I have a set of independent random walkers on a graph that produce a distribution of the nodes in that graph. I would like to test whether the random walkers have produced a distribution which is consistent.

I thought about separating the results from the random walker into various sets, eg 4 separate groups, and using the distribution that these 4 iid groups make from the graph, use Gelman-Rubin diagnostics link to def.to test for convergence of those distributions.

I cannot use Gelman and Rubin's potential scale reduction factor directly on the independent walkers, because they are not let to run till they reach ergodic steady states. I suppose that the reasoning behind using Gelman and Rubin still holds when comparing within $W$, and between variance $R$ of distributions from groups of these independent walkers because even a markov chain run till it is supposed to reached ergodicity is a distribution.

Is my approach valid and why? are there any references for it?

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When performing Markov chain Monte Carlo, you never know when and if the ergodic steady state is reached. Gelman-Rubin statistic is a test for reaching this state.

An important assumption for the Gelman-Rubin statistic is overdispersion of the initial states. This is needed to guarantee that the between-chain variance approaches the variance of the posterior from above. Make sure that this assumption is satisfied.

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    $\begingroup$ can the use of multiple iid walkers be used in aggregation to approximate that state. The search is for testing the locality of a discrete distribution's consistency for the iid walkers $\endgroup$ – Vass Sep 2 '13 at 10:47

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