Answers:
看来您已经放弃了共轭。仅作记录,我见过人们正在做的一件事(但不记得确切的位置,对不起)是这样的重新参数化。如果是条件IID,给定的α ,β,使得X 我 | α ,β 〜乙Ë 吨一个(α ,β ),记住 È [ X 我 | α ,β ] = α 和 V一- [R[X我|α,β]=αβ
是的,它在指数族中有一个共轭先验。考虑三个参数族 为的某些值(一个,b,p),这是积的,虽然我还没有完全想出其中(I相信p≥0和一个<0,b<0应该工作-p=0对应于独立指数分布,从而绝对有效,共轭更新涉及增量
该问题,并且在的原因至少一部分没有人使用它,是 即归一化常数没有封闭形式。
In theory there should be a conjugate prior for the beta distribution. This is because
However the derivation looks difficult, and to quote A Bouchard-Cote's Exponential Families and Conjugate Priors
An important observation to make is that this recipe does not always yields a conjugate prior that is computationally tractable.
Consistent with this, there is no prior for the Beta distribution in D Fink's A Compendium of Conjugate Priors.
我不相信有一个“标准”(即指数族)分布是β分布之前的共轭物。但是,如果确实存在,则必须是双变量分布。
Robert and Casella (RC) happen to describe the family of conjugate priors of the beta distribution in Example 3.6 (p 71 - 75) of their book, Introducing Monte Carlo Methods in R, Springer, 2010. However, they quote the result without citing a source.
Added in response to gung's request for details. RC state that for distribution , the conjugate prior is "... of the form
where are hyperparameters, since the posterior is then equal to
The remainder of the example concerns importance sampling from in order to compute the marginal likelihood of .