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Inspired by this question, and particular "Problem 3":

Posterior distributions are somewhat more difficult to incorporate into a meta-analysis, unless a frequentist, parametric description of the distribution has been provided.

I've been thinking a great deal recently about incorporating meta-analysis into a Bayesian model - primarily as a source of priors - but how to go about it the other direction? If Bayesian analysis does indeed become more popular, and becomes very easy to incorporate into existing code (the BAYES statement in SAS 9.2 and above comes to mind), we should more frequently get Bayesian estimates of effect in the literature.

Let's pretend for a moment that we have an applied researcher who has decided to run a Bayesian analysis. Using the same simulation code I used for this question, if they went with a frequentist framework, they'd the following frequentist estimates:

log relative risk = 1.1009, standard error = 0.0319, log 95% CI = 1.0384, 1.1633

Using a standard, all-default and uninformative priors BAYES statement analysis, there's no reason to have a nice, symmetric confidence intervals or standard errors. In this case the posterior is pretty easily described by a normal distribution, so one could just describe it as such and be "close enough", but what happens if someone reports a Bayesian effect estimate and an asymmetrical credible interval? Is there a straightforward way to include that in a standard meta-analysis, or does the estimate need to be shoehorned back into a parametrically described distribution that's as close as possible? Or something else?

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    $\begingroup$ There's also the additional issue that if they incorporate non-weak prior information the meta analysis should try to avoid double counting that information from multiple studies that use the same prior information. $\endgroup$
    – John Salvatier
    Oct 27, 2011 at 4:30
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    $\begingroup$ Maybe, starting with the first study, and iterating - with each studies' posterior becoming the prior for the next. So what if the intervals are skewed - are we talking about publish-ability? The resulting "curve" of the changing distributions over time would give you information about the development of the field too. Would there be a good way to look at publication bias though? Perhaps a type of control-chart - where too many successive "positive" results would be detected. $\endgroup$
    – user6666
    Oct 29, 2011 at 12:54

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Something else. To perform Bayesian analysis on the results of several studies that address the same parameter (or parameters) you need to get hold of their likelihoods - or approximations thereof - and multiply them by the prior.

If each individual analysis has reported its just its own Bayesian inference, this will not be possible - though an approximation might be feasible. Happily, most papers will report a straight summary of the data before giving their fully Bayesian inference. For your Bayesian inference, you can start with that summary and add your prior.

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