For which distributions x, other than beta, is the x-binomial distribution nice? The beta and binomial distributions are famously conjugate but I am curious if other non-conjugate distributions will give comparably simple compound pmfs. By nice I mean that the pmf is nice to compute without resorting to numerical integration; I'm not referring to the easyness of sampling from the compound distribution.
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Nice question. I was going to suggest a Jeffreys prior, but it turns out that's just [a special case of the beta distribution for binomial models](http://www.cs.berkeley.edu/~jordan/courses/260-spring10/lectures/lecture7.pdf). I'll keep thinking and see if I can come up with any others. I have some vague sense that there are transformations of Gaussians that work, but Google isn't helping me pin that down. – David J. Harris Aug 22 '12 at 22:14
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If you just want to be able to write down the probability mass function, you have a lot of flexibility, basically because you can repeatedly use integration by parts. As long as you can integrate the distribution of $X$ repeatedly to get closed form expressions, you get at worst a double sum of closed form expressions for the pmf of the compound distribution. I think you often get single sums, so maybe there is an even simpler way of expressing these.
For example, let $\text{pdf}_X(x) = -\log x$ on $[0,1]$.
$$ \int_0^1 {N\choose k}x^k (1-x)^{N-k} (-\log x) ~dx \\\ = \frac{1}{N+1}\sum_{i=k}^{N} \frac1{i+1}.$$
Douglas Zare
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