let's say theta is modeled by a Gaussian
The conjugate prior of the binomial distribution is the beta distribution, so if you use a beta distribution for theta, the posterior is also a beta distribution, and the expected value of the posterior predictive is just (u0 + u)/(u0 + u + d0 + d) where u and d are the number of up- and downvotes and u0 and d0 are the parameters of the prior distribution, or pseudocounts.
You're right, that's in the second chapter of Gelman too. I'll edit that.
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