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.
Note that when u0 and d0 are zero, or negligible because the total number of votes is large, your posterior expectation is just u/(u+d) -- in other words, exactly the %positive that LW reports when you hover over the score.
(But in practice the total number of votes is rarely large, so the prior matters.)
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.
Note that when u0 and d0 are zero, or negligible because the total number of votes is large, your posterior expectation is just u/(u+d) -- in other words, exactly the %positive that LW reports when you hover over the score.
(But in practice the total number of votes is rarely large, so the prior matters.)