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.
