Hi
I am wondering if there is a standard effective sample size (ESS) to declare the convergence of a chain. Well, the convergence of a MCMC chain is obviously assessed using different convergent diagnostic tests: Gelma-Rubin, Gewke, Heidelberger-Welch test... However, if I would evaluate the ESS of parameters estimated by Gibb sampling how to define the values are adequate.
Example. My samples were 10,000, after burn-in, thin... The ESS for estimates are
Var1= 780
Var2= 323
Var3= 2,963
Var4= 5,456
Var5= 174
Var6= 8,351
Are ESSs of Var1, Var3 and Var4 acceptable? Should I run again a MCMC?
Thanks in advance for your answers.
I do not see sample size as guaranteeing convergence, I see ESS as indicating you are being more or less effective in characterising the posterior. That is ESS is useful for "Inference from Simulations after Approximate Convergence ". Consequently, it depends a great deal on what you are doing and the inference you want to make; if you just want a point estimate you are going to need not that many; in teaching I advise 500 after the burnin-n and convergence appears to be achieved; but if you want 95% credible intervals around that point estimate, then you will need a more . In many models the computational cost (especially in relation to the cost of collecting the data) is trivial.
see Bayesian Chapter in this Book Developing multilevel models for analysing contextuality, he...
and
http://www.stat.columbia.edu/~gelman/research/published/GelmanMCMCchapter4.pdf
and
see Article Multivariate Output Analysis for Markov chain Monte Carlo
For MCMC, convergence can be traced by using the chains plot after specifying the burn in, let's say 200, and simulating the sample from small to large, let's say 20 samples to 20,000 samples. The more the chains, the higher the convergence, and hence stationarity of the variables
The more the chains, the higher the convergence, and hence stationarity of the variables
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