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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?

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  • $\begingroup$ Have you tried MLE? $\endgroup$ – Tony Hellmuth May 29 '18 at 2:10
  • $\begingroup$ Do you have a closed form solution at hand? I don't have the chops to know if one even exists. $\endgroup$ – zipzapboing May 29 '18 at 2:15
  • $\begingroup$ 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. $\endgroup$ – Tony Hellmuth May 29 '18 at 2:16
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    $\begingroup$ 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/… $\endgroup$ – Tony Hellmuth May 29 '18 at 2:26
  • $\begingroup$ 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. $\endgroup$ – zipzapboing May 29 '18 at 2:34
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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.

Sourced from "A First Course in Bayesian Statistical Methods"

Sourced from "A First Course in Bayesian Statistical Methods"

Sourced from "A First Course in Bayesian Statistical Methods"

Sourced from "A First Course in Bayesian Statistical Methods"

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