Short question: What happens to the beta-binomial distribution, when n increases to infinity? Is there a count distribution arising like it's for the classical binomial distribution?


Consider the urn model for the beta-binomial:

... imagine an urn containing α red balls and β black balls, where random draws are made. If a red ball is observed, then two red balls are returned to the urn. Likewise, if a black ball is drawn, it is replaced and another black ball is added to the urn. If this is repeated n times, then the probability of observing k red balls follows a beta-binomial distribution with parameters n,α and β.

When both $\alpha$ and $\beta$ are very large, the additional balls introduced do not appreciably change the chances of red or black. This is just a binomial experiment (with fixed probabilities $\alpha / (\alpha + \beta)$ of red and $\beta/ (\alpha + \beta)$ of black).

  • $\begingroup$ I think $\alpha, \beta$ are fixed and he's sending $n\rightarrow\infty$ $\endgroup$ – JMS Jun 14 '11 at 17:02
  • $\begingroup$ @JMS If that's the case, you recover the underlying Beta distribution if you divide by $n$. $\endgroup$ – whuber Jun 14 '11 at 18:12
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    $\begingroup$ Right; without additional scaling the beta-binomial won't converge to a proper probability distribution as $n\rightarrow\infty$ if $\alpha, \beta$ are fixed (I think). From his phrasing ("count distribution") I think the OP may be looking for the analogue of a binomial approaching a Poisson as $n\rightarrow \infty$ if $p\rightarrow 0$ and $np$ is constant. Probably there is a similar result if you send $\alpha/(\alpha+\beta)$ to zero at a proper rate or something, but I don't know it. $\endgroup$ – JMS Jun 14 '11 at 20:29

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