Using variance from a Jeffreys prior in a Prevalence meta-analysis I am doing a meta-analysis on a group of studies, where the observation is a rate (no. of successes / no of trials). Some of the studies have small sample sizes (n<10) and/or did not observe any successes. Because of this I did not do the typical normal approximation SE calculation on the proportion for inputting into the meta-analysis. 
As an alternative I used Jeffreys Priors and calculated 95% Bayesian Credible intervals for each rate. 
Then in stata using the metan function, I inputted the observed rate and two bayesian intervals. The function runs fine of course and I can generate estimates, but was curious if this is appropriate to do. I believe stata takes the confidence intervals and backwards calculates the SE/variance and uses this in weighting. 
I was curious if anyone thought this is inappropriate to use bayesian intervals in the mixed effects Meta-analysis function?
 A: Suppose you really want to go the route with Jeffreys prior. So, in a particular study, you have $y$ cases/successes out of $n$ trials. Assume that the random variable $y$ follows a Binomial distribution, so that: $$f(y,\pi) = {n \choose x} \pi^y (1-\pi)^{n-y},$$ where $\pi$ is the true probability of a success.
Now let's place Jeffreys' prior on $\pi$. It turns out that Jeffreys' prior in that case is the Beta distribution with parameters $a=1/2$ and $b=1/2$. And since the Beta distribution is conjugate in this case, the posterior is easy to obtain: It is Beta with parameters $\alpha = y + 1/2$ and $\beta = n - y + 1/2$.
Therefore, the posterior mean is $$E[\pi|y,a,b] = \frac{y + 1/2}{n + 1}$$ while the posterior variance is $$Var[\pi|y,a,b] = \frac{(y + 1/2)(n - y + 1/2)}{n(n+1)^2}.$$
So, you could compute these values for each study and feed them to the meta-analysis software of your choice (or take the square-root of the posterior variances if the software is asking you to supply the standard errors). However, this doesn't get your around the problem of normality. The usual inverse-variance method (or 'normal-normal' model) is based on the assumption that the sampling distributions are (at least approximately) normal. In this example, we know that the distributions are not normal, but in fact Beta. If the posterior mean is not too close to 0 or 1, approximating the posterior by a normal may be fine, but certainly not when you observe no or only very few successes.
For illustration, assume $n=10$ and let's examine what those posterior distributions look like for various values of $y$:
n <- 10
par(mfrow=c(3,3), mar=c(3,4,3,1))
for (y in c(0,1,2,3,5,7,8,9,10)) {
   hist(rbeta(100000, y+1/2, n-y+1/2), col="lightgray", xlab="", main=paste0("y = ", y), xlim=c(0,1), freq=FALSE)
   curve(dnorm(x, mean=(y+1/2)/(n+1), sd=sqrt((y+1/2)*(n-y+1/2)/(n*(n+1)^2))), from=0, to=1, add=TRUE, col="red", lwd=2)
}


So, unless $y$ is close to 5, the normal approximation doesn't work. So, using the binomial-normal model would probably be the better approach here.
