# MCMC chains and Convergence

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,

• 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. – Xi'an Apr 12 at 10:00