# Log Beta Distribution Priors [closed]

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:  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

• I am curious how your dataset can indicate what the prior distribution ought to be. Could you explain? – whuber Oct 9 '18 at 18:41
• @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. – Alex R. Oct 9 '18 at 18:48
• But how do you obtain an empirical distribution for $\theta$? Are you performing this estimate repeatedly in many similar circumstances? – whuber Oct 9 '18 at 18:53
• @whuber: Right, I have a large dataset of triples $(\theta,S,F)$. – Alex R. Oct 9 '18 at 18:54
• 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. – whuber Oct 9 '18 at 18:59