Hi,
I'm glad you think the tree estimates are interesting!
I think the obvious thing to do is to check that ASDSF < 0.01
and that the ESS values are large enough. The ASDSF checks that
the tree topologies are converging, and the ESS checks that the
parameter estimates are converging. There is not really a magic
number for the ESS, but maybe all the ESS numbers should be >
500. Probably this gives similar results to looking at the ASDSF
and the PSRF.
However, I think the ultimate goal is not to follow some simple rules, but rather to convince yourself that (i) the multiple different chains are converging to a common equilbrium, (ii) the results will not change if you run it longer, and that (iii) if you run the analysis a second time, your results should be similar.
In order to convince yourself that the multiple chains are
converging, all the different diagnostics are helpful, but you
don't have to use all of them. If you are in doubt, you could
always run the chains a bit longer and see if anything changes, or
run the analysis again and see if you get the same results.
Does that help?
-BenRI
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