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I have results from 25 experiments that include an estimate of mean effect and a thousand posterior samples of the effect.

Rather than use the standard deviation of these posterior samples as the standard error, and then assume a normal prior, can I directly incorporate these 1000 samples for each experiment?

My question is related to So how would you include Bayesian estimates in a meta-analysis?, but I don't think the answer directly addresses this question.

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I will assume that these are analyses without much prior information incorporated (I think there might be issues, if all used the same relatively strong prior information, that would then get counted repeatedly), and most importantly that results from one experiment were not then used to set a prior for the next one. If that's the case, then the most obvious solution to me is to find a distribution that approximates these samples. Often, after some suitable transformation, something like a normal, Student-t or skew-t should provide a pretty decent fit, but you can also do some mixture distribution, which with enough mixture components can approximate any smooth distribution closely. You can then use that as your trial level likelihood. That may be a bit more tricky in a frequentist meta-analysis, but is pretty straightforward in a Bayesian meta-analysis.

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  • $\begingroup$ Yes - just that I will have to first fit a mixture distribution for each set of posteriors. I wish there were an automatic empirical distribution sampler in STAN or pymc3. Or perhaps I can treat the model each sample as a repeated measure... $\endgroup$
    – pythOnometrist
    Jun 8, 2022 at 23:57
  • $\begingroup$ For fitting a mixture, e.g. the RBesT R package is great, but yeah, it's a bit of a detour. I don't think these samplers could work when given samples instead of a likelihood, because they need differentiable functions. Same reason why discrete parameters are so tricky. $\endgroup$
    – Björn
    Jun 9, 2022 at 5:23

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