# How to parametrize a beta prior for a sequential Bayesian analysis of a proportion?

I'm developing a decision rule for a study using an interim analysis with a Bayesian characterization of the event probability. At a certain point, we will perform a planned stop, and only proceed with the study if it meets certain benchmarks. I'm interested in ensuring that the probability that the posterior exceeds a threshold is $$\beta = 0.8$$ or higher. This is implemented in the R package ph2bayes, but the citations and literature don't get into the nitty gritty of using an informative prior.

Suppose at my planned interim, I have 0 events out of 5 observations, and I will enroll 20 if I meet the "go" criterion, and I would like the posterior probability of the proportion being 30% or higher to be 80%. How can I use 0 and 5 to define the shape and scale parameters of a beta distribution to describe binomial likelihood of the proportion as a prior?

What you are looking for is a set of prior parameters $$(a, b)$$ such that the posterior distribution given $$0$$ events out of $$5$$ observations gives an 80% probability that the true binomial probability parameter $$>0.3$$. Unfortunately, as you have one criterion and two parameters, this isn't an identifiable problem - there are multiple parameterizations that will get you that result. You can fix this by adding another criterion, and I will suggest a couple below.

To see this, look at the posterior, which is a $$\mathrm{Beta}(a+0,b+5)$$ distribution. We need:

$$\int_0^{0.3}p(x;a,b+5)\mathrm{d}x = 0.2$$

A little root-finding gives us the parameter combinations $$(7.022, 5.552)$$, $$(5.506, 2.748)$$, etc.

We might try to use the least informative, in some sense of the word ("equivalent sample size" in this case) prior, the one which has the minimum of $$a+b$$. A little thought will convince us that this will occur when $$b=0$$, as $$a$$ must be an increasing function of $$b$$ in order to preserve the quantile value. This is an improper prior, but since we know we will add $$5$$ to the value of $$b$$, we know the posterior will be proper. We calculate the corresponding value of $$a$$ as follows:

uniroot(function(a) pbeta(0.3, a, 5) - 0.2, lower=0.001, upper=10)$root [1] 3.953105  The CDF does not support the "minimum equivalent sample size" approach to a minimum information prior, with roughly $$97\%$$ of the mass at values $$> 0.99$$, which most of us would think of as highly informative: An alternative might be to fix $$b=1$$, with the weak thought that $$a=1, b=1$$ define a uniform (Laplace) prior on the probability: uniroot(function(a) pbeta(0.3, a, 6) - 0.2, lower=0.001, upper=10)$root
[1] 4.528547


Our equivalent sample size is $$5.53$$, not much more than the previous $$3.95$$, and the CDF looks much better, with roughly $$95.6\%$$ of the mass at values $$< 0.99$$:

Clearly, there is going to be some unavoidable subjectivity in the choice of prior.

• Does the b component of the conjugate beta posterior add the $n$ or the $y-n$? Commented Jun 2, 2022 at 22:23
• You have 0 successes, which add to the $a$, and 5 failures, which add to the $b$, so that's a yes. Commented Jun 2, 2022 at 22:37