I'm modelling count variables with a fixed number of trials, hence a binomial problem. The data is overdispersed, and in this case, overdispersion plays a crucial role. In the past, I fitted a beta-binomial distribution via ML.
Now, I want to turn to a Bayesian estimation. What are appropriate uninformative or weakly informative priors for the parameters a and b of the beta-binomial distribution? Note that I have no predictors, just the plain null model.
The beta distribution is sort of annoying to deal with; I would avoid it if I were you, in favor of a logit or probit model. You can then put weakly informative normal priors on the transformed log-odds/probit parameters. The normal prior on log-odds is more intuitive than trying to build a joint prior over a and b in the beta, and is a standard approach.
Hi,
I use to reparameterize:
x~Beta(a,b)
a=mu*eta
b=mu*(1-eta)
Then
E(x)=mu
V(x)=mu*(1-mu)/(eta+1)
Prior for mu could be another Beta and eta might have a Gamma or log-normal prior.
I normally do not resort to putting a prior on the beta distribution, unless it is multi-task (population) problem and I need a separate beta distribution for each dataset, with the assumption that each dataset's beta distribution was drawn from the same prior. Then I simply use an exponential prior for alpha, beta. If computation is not an issue, then this admits a Gibbs sampler if I remember correctly.
If this is just one set of data, then Richard's suggestion is great. It's also very good for fitting a linear-Gaussian model on the transformed data (which is an approach a friend of mine used recently).
Martin,
Use your data in the binomial likelihood, and then use as a prior a Beta(0.5,0.5). That Beta is the classic Jeffrey's prior for any Bernoulli based likelihood, as well as the classical ME prior, and reference prior. By the way, the result is another Beta model, so in this case you get a nice closed form analytical solution.
Mark Powell
The typical prior for the beta density's parameters are gamma densities. I usually set the gamma parameters such that themean of the gamma prior is 1 with a moderately large variance,eg. ~10. An alternative is the lognormal. Another is the uniform density from 0 to a small integer, eg, 10.
Thank you everybody. Seemingly, there are multiple options:
1) Avoid beta-binomial in favor of transformed log odds. (Richard)
2) Re-parametrization (Samu)
3) Exponential or gamma priors (Christos, Joseph)
@Mark and Christos: the point is, that the beta prior reflects a true random effect, and the size of the random effect (i.e., the amount of overdispersion) is central to the prediction. Therefore, I have to use priors on the beta parameters. Btw., the primary reason I was asking this question, is that when you search Google for priors beta, all results are about the beta as a conjugate prior.
I am glad to see that Richards suggestion got so many votes, because this is what i did already, modeling the random effect by a logit-normal distribution. Apparently, my intuition was wrong that a beta-binomial would be better suited. Now, I just have to figure out how to implement the logit-normal in Stan or BUGS. Anybody?
@Martin OK, so if the problem is just overdispersion then I wouldn't bother with anything other than logit, even if it strictly speaking it is not the correct model. It _should_ be easy to do it with BUGS. If you do implement it, maybe you can share the model description :)
Martin, I've attached a working example. For those that don't want to open the file, the BUGS code is:
model{
for(i in 1:N){
y[i] ~ dbin(p[i],M[i])
logit(p[i])
(You know, until now I never thought that researchgate forums could be useful)
If the Beta-layer of the Beta-Binomial distribution is not of interest, you might want to still use the Beta-binomial, because the random effects have been integrated out. As a result you will have better behaving MCMC.
In BUGS you can use the "zeroes trick" to implement the Beta-bin likelihood. And in JAGS Beta-binomial is nowadays a built-in distribution.
@Richard: Thanks for the BUGS code. I hoped it would be as simple as just adding another layer of priors. Has someone not yet heard of Stan? Here is a tutorial on how logistic regression is connected to binomial data [1], Stan code included.
What I’m trying to do is to estimate the completeness of intellectual discovery processes, such as
• spell checking
• expert reviews
• theme identification in qualitative research
• requirements elicitation (software engineering)
• Usability validation of medical devices
For an illustration, imagine the situation of multiple experts reviewing your latest paper before submission. Your paper has a number of weak points. The more you can iron out before submission, the better your chances for a minor revision. Let’s assume the experts have about the same skills. What differs is the visibility of the weaknesses. The number of weaknesses is fixed but unknown. After receiving, let’s say, 5 reviews, you have discovered 30 weaknesses. Naturally, you also want to know, how many weaknesses remain undiscovered in your work, limiting its quality.
What the above situations have in common, is that you choose your sample size to get to a desired level of completeness. A discovery process is complete when you have observed all items at least once, y>0.
Imagine what happens if visibility of items varies. The observed frequency distribution becomes more spread out, frequency at y=0 increases. This is the reason why the beta prior matters, it is a random effect that matters.
You may also want to express your belief in the estimated completion, using credibility intervals. This brings us directly to Bayesian modeling. I’ve implemented the model using the likelihood. Credibility of estimated completion I did with bootstrapping.
It’s natural to step over to Bayesian modeling, because there is belief: discovery processes are often repeated. For example, it is not the first time that experts review your paper before submission. In the long term, through shrinkage, you get the desired credibility earlier.
In summary, we have three complications:
1. Data is zero truncated (or positive). Items with zero success are never observed
2. Two levels of priors are needed, on top of the zero-truncated data:
- empirical prior: the logit-normal, approximating the visibility variance of items
- belief priors: mean and variance of the logit-normal
3. How can we model previous belief as a continuous meta study(?)
[1] http://lingpipe-blog.com/2012/10/30/upgrading-from-beta-binomial-to-logistic-regression/
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