This is the third of a series of articles about Bayesian analysis. The previous article is here.
Earlier this semester I posed this problem to my Bayesian statistics class at Olin College:
This is an example of a mark and recapture experiment, which you can read about on Wikipedia. The Wikipedia page also includes the photo of a tagged hyrax shown above.Suppose I capture and tag 10 rock hyraxes. Some time later, I capture another 10 hyraxes and find that two of them are already tagged. How many hyraxes are there in this environment?
As always with problems like this, we have to make some modeling assumptions.
1) For simplicity, you can assume that the environment is reasonably isolated, so the number of hyraxes does not change between observations.
2) And you can assume that each hyrax is equally likely to be captured during each phase of the experiment, regardless of whether it has been tagged. In reality, it is possible that tagged animals would avoid traps in the future, or possible that the same behavior that got them caught the first time makes them more likely to be caught again. But let's start simple.
My solution to this problem uses the computation framework from my book, Think Bayes. The framework is described in this notebook. If you have read Think Bayes or attended one of my workshops, you might want to attempt this problem before you look at my solution.
If you solve this problem analytically, or use MCMC, and you want to share your solution, please let me know and I will post it here.
And when you are ready, you can see my solution in this notebook.
I will post more of the exercises from my class over the next few weeks.
UPDATE December 5, 2014: João Neto posted a solution to this problem in BUGS using a Jeffrey's prior.