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I'm a newbie in Bayesian modelling and trying to understand something more on such field by running a Poisson regression and analyzing count data.

Browsing on the internet, I found a set of diagnostics for assessing the model convergence; among all of these ones, I read about the Geweke's diagnostics.

I know it rejects the hypothesis of convergence for large z-score, but I wonder of how much the z-score has to be large to reject the hypothesis. There exists a rule of thumb?

Moreover, in the case one has a really large z-score, let's say 3, what you could do? I read about one has to have a longer chain, but I really did not get what does it mean.

Could you provide some help, possibly by providing a reference about?

Thanks in advance.

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MCMC provides the user with a procedure to obtain samples from some target distribution (typically a posterior distribution in the Bayesian framework). The way the procedure works is that the distribution from which samples are obtained from converges to the target distribution.

Convergence diagnostics are required to assess when the sampling procedure has converged to sampling from the target distribution.

If the diagnostic indicates that the sampler process has not converged to sampling from the target distribution MORE samples will need to be drawn (and/or more of the initial samples drawn should be discarded (burnin-in) as they are not from the target distribution)

The Geweke diagnostic is pretty simple. It compares the mean of the samples drawn from the end of a chain of MCMC output to the mean of the samples at the beginning of the chain using a $z$-test like statistic. The variance of the two means is computed using spectral densities (b/c the MCMC output is not independent).

As for a cut-off you can compare to the standard normal critical values $z_{\alpha/2}$ where $\alpha=0.05$ or something similar.

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