Let $X_n$ be a binomial distribution with parameter $\theta$. Empirically, after $n$ throws, my estimate is $\hat{\theta}_n=\frac{S_n}{S_n+F_n}$, where $S/F$ are the successes and failures $(F_n:=n-S_n)$.

From simulations on my dataset, I've found out that $-\log(\theta)$ is better approximated by a beta distribution, as opposed to the usual way of assuming $\theta$ has beta binomial prior $\mbox{Beta}(a+S_i,b+n-S_i)$. In other words, let $\phi_n:=\frac{-\log(\theta_n)-c}{d}$, ($c,d$ are shift/scaling constants) so that $\phi_n$ is assumed to have empirical distribution $\mbox{Beta}(a,b)$. Here's a qq-plot comparison, with $\theta$ assumed to be beta in the first image, and $-\log(\theta)$ assumed to be beta in the second:

Theta Assumed to be Beta

-Log(Theta) Assumed to be Beta

Question 1: Are there any good examples of real-life situations where the success probability is log-beta distributed? The literature search on this is extremely annoying because it seems like people call it either log-beta or exp-beta (I'm following the log-normal convention).

Question 2: Is there any way of easily calculating the posterior update for the log-beta distribution, based solely on $S_n$ and $F_n$?. $\log(\theta_n)=\log(S_n)-\log(S_n+F_n)$, but this no longer has the interpretation of success/failure. The integral looks to be a sum of confluent hypergeometric functions

  • $\begingroup$ I am curious how your dataset can indicate what the prior distribution ought to be. Could you explain? $\endgroup$ – whuber Oct 9 '18 at 18:41
  • $\begingroup$ @whuber: I'm taking it as only empirical evidence. It also does a better job than a few other families that I've tried. That's why I'm interested in Question 1, because it would be compelling to at least see this happening elsewhere or have an example of a probabilistic process that gives rise to a log beta distribution. $\endgroup$ – Alex R. Oct 9 '18 at 18:48
  • $\begingroup$ But how do you obtain an empirical distribution for $\theta$? Are you performing this estimate repeatedly in many similar circumstances? $\endgroup$ – whuber Oct 9 '18 at 18:53
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    $\begingroup$ @whuber: Right, I have a large dataset of triples $(\theta,S,F)$. $\endgroup$ – Alex R. Oct 9 '18 at 18:54
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    $\begingroup$ You might search for situations where the logit of the parameter has a given distribution. For the small values you report, the logit and log are nearly the same. This would point you towards Bayes versions of logistic regression. $\endgroup$ – whuber Oct 9 '18 at 18:59