# Sufficient statistics for beta-binomial distribution

The beta-binomial pmf is

$$f_X(x) = {n \choose x}{B(x+\alpha, n-x+\beta)\over B(\alpha, \beta)}$$

where $B$ is the beta function.

The numerator is the issue here when trying to separate data from parameters. I tried using

$$B(x+\alpha, n-x+\beta) \propto \Gamma(y+\alpha)\Gamma(n-x+\beta)$$

and the fact that $x$ is discrete but that led me to a polynomial where parameters and data are still intertwined through the exponents and coefficients.

Is this solvable at all?

• Have you tried MLE? – Tony Hellmuth May 29 '18 at 2:10
• Do you have a closed form solution at hand? I don't have the chops to know if one even exists. – zipzapboing May 29 '18 at 2:15
• I don't, but I shall try searching. Perhaps there are no sufficient statistics. Although by inspection I feel it will be something like the sum of logs. Also I assume we are doing Bayesian analysis here. – Tony Hellmuth May 29 '18 at 2:16
• Hmm I think $\sum_iy_i$ is sufficient. Given the average, it should mean we can get the distribution of our posterior. See if this helps: www1.se.cuhk.edu.hk/~seem5680/lecture/… – Tony Hellmuth May 29 '18 at 2:26
• That was my first intuition, but I was hoping I could provide more traditional proof. I was not familiar with the Bayesian approach to sufficiency... my Bayesian Inference class has been nothing but writing Gibbs samplers for trivial models all semester long. I feel quite cheated. Anyway! If you post this as an answer I will accept it. – zipzapboing May 29 '18 at 2:34

Here, we have $p(y|\theta)$ is our data from a Binomial distribution, parameters $n,p$. And $p(\theta)$ is from a Beta distribution, parameters $a=1,b=1$ for simplicity. The final image shows that the hyper parameters are just additive, where $dbeta$ is essentially the Binom-Beta distribution.