# Estimating the binomial distribution parameter $p$ for rare events

Suppose we have an extremely large collection of red balls and green balls. If we let $$R$$ and $$G$$ be the events of drawing a red ball and drawing a green ball, respectively, and if we let $$\Pr(R)=p$$ then $$\Pr(G)=1-p$$. We also assume that the vast majority of the balls in our collection are green, implying $$p<<1-p$$.

We would like to estimate $$p$$ using the following experiment: sample from the collection 3 times by selecting 20 balls each time. Then starting with the uninformed prior distribution of $$p$$ as Beta$$(1,1)$$ we update the prior by Bayesian updating after each sample and compute the expected value of $$p$$ after completing the 3 samples.

So suppose we sample and get 20 green balls for each of the 3 samples. The final update of the prior distribution of $$p$$ is Beta$$(1,61)$$ and the expected value of $$p$$ is $$E(p)=\frac{1}{62}\approx 0.01613$$.

This seems to be a very high estimate and I think the problem is that I am starting the updating with an uninformed prior despite the fact that I know $$p<< 1-p$$. Is there a way to justify a more informed prior in order to get a stronger conclusion? For example, if I believe that $$p\le 0.01$$ start with the informed prior Beta$$(2,100)$$, my update for $$p$$ will be distributed as Beta$$(2,160)$$ and $$E(p) = \frac{1}{81}>0.01$$! That makes no sense to me. Any help would be appreciated.

Seeing 0/60 isn't strong evidence for $$p\leq 0.01$$, since seeing 0/60 is pretty much what you expect for $$p\leq 1/60$$. This means your posterior won't have high weight on $$p\leq 0.01$$ unless your prior does
If you are confident a priori that $$p\leq 0.01$$ you want the prior probability of that to be high. A Beta(2,100) prior still only has 26% probability of $$p\leq 0.01$$; the Beta(2,160) prior has about 48% probability, which seems plausible. Note that the prior and posterior are very skewed, so although the posterior median is close to 0.01, the posterior mean is higher.
If you had a Beta(2,200) prior, the prior mean would be 0.01 and the prior median a bit lower. The posterior mean would be about .0075, the posterior median about 0.0065, and the posterior probability of $$p\leq 0.01$$ about 75%
So, there's two things going on: the data don't provide much evidence that $$p$$ is very small rather that just small, and your prior is much weaker than what you describe as your actual beliefs.
• Thank you! Your comments make perfect sense. If I solve $(1-p)^{60}=0.5$ for $p$, then that tells me I have a $50\%$ chance that $p\approx 0.011486$ based on the test results, so starting with a prior distribution of Beta$(2,160)$ is reasonable. Then the posterior is distributed as Beta$(2,220)$ so that $E(p)=\frac{1}{111}$. Commented Oct 24, 2020 at 5:19