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This may be a dumb question to ask, but I was doing MCMC sampling of my parameters $\alpha_1, \ \alpha_2 .... \alpha_{50}$, where $\alpha_i$ denotes for the same parameter $\alpha$ at the time i.

I computed Gelman-Rubin Statistics for my MCMC samples and interestingly, the GR-Statistics show that the convergence of the parameters $ \alpha_1, \ \alpha_2 .... \alpha_{49}$ are considered to be poor, but the convergence of the $\alpha_{50}$ chain is excellent. Since the prime reason for my MCMC sampling is to make a prediction based on $\alpha_{50}$, I am wondering if I should be concerned that the convergence of the chains for $ \alpha_1, \ \alpha_2 .... \alpha_{49}$ are not too great? or can I just take the fact that the chain for $\alpha_{50}$ has converged, and just go ahead and make predictions based on $\alpha_{50}$?

Thank you,

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    $\begingroup$ Markov chains produced by MCMC may converge at a different rate according to the component one considers. So there is no theoretical issue with one component $\alpha_{50}$ apparently converging faster than the others. Note however that the Gelman-Rubin criterion is not fool-proof, i.e., may indicate convergence despite some posterior mass missing from the simulation. I would advise running a checkup with simulated data where the values of the $\alpha_i$'s are known, so that one can check the coverage of these values by the MCMC samples. $\endgroup$ – Xi'an Apr 12 at 10:00

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