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I use Gelman-Rubin statistics, trace plots, autocorrelation plots and effective sample size to check the convergence. However, I got very different results from the above tests. The Gelman-Rubin statistics are below 1.05 but the effective sample size is quite small, only 200 for each parameter (burn-in 50000, iter 100000). And the autocorrelation plot shows the parameters are highly correlated. I tried to thin the chains but the result is worse. I have the following questions:

  1. What should I do to improve the effective sample size?
  2. Can I neglect the effective sample size and conclude the chains have converged?
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  • $\begingroup$ Gelman-Rubin and ESS are connected: read full description of the connect here: arxiv.org/abs/1812.09384. In gist, the R-hat less than 1.1 is arbitrary, and too liberal. To improve effective sample size, run the sampler for far more steps. $\endgroup$ – Greenparker Apr 8 at 9:38
  • $\begingroup$ @Greenparker Actually, the GR statistics are below 1.05 for all parameters. But thanks for your link. $\endgroup$ – Suki Hao Apr 8 at 9:43
  • $\begingroup$ Are you sure your proposals are sensible and do not get rejected most of the time? $\endgroup$ – Tom Apr 8 at 9:56
  • $\begingroup$ @Tom I am using Bayesian Poisson model with a log link, so there is no proposal. $\endgroup$ – Suki Hao Apr 8 at 10:14
  • $\begingroup$ @SukiHao Below 1.05 is also too liberal as pointed out in our paper. After burn-in, you get 200 ess from 50,000 samples. If you want 10,000 ess then you'll need approximately 2.5 million samples. This seems reasonable, considering your Markov chain seems likely to be slowly mixing. $\endgroup$ – Greenparker Apr 8 at 10:49

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