Bayesian prior: What should I set the beta parameters to when we know that proportion estimate is not 50%?

Question

When trying to estimate the proportion of a sample (eg. proportion of patients undergoing an operation who experience a complication) one starts with the data (number of "successes", number of trials) and a prior distribution of that proportion parameter, typically beta(1,1).

What happens when one knows as sure "as God made little apples" that the proportion who will experience complications is not 50%. What happens if we are confident that the parameter possibilities do not span the spectrum from 0 to 100%. We expect the complication rate to be about 10% or there abouts. 50% would be very very very unlikely. If you want to see more details about my case you can look here but my question is a general one.

Answer 1

"We expect the complication rate to be about 10% or there abouts. 50% would be very very very unlikely."

These two sentence can be translated into a prior. As you point out, the Beta distribution is the conjugate prior for the Binomial, so that's what we'll use. If the complication rate is about $10\%$, then it makes sense to use a beta distribution with a mean of around $0.1$. All we need to do is pick the $\alpha$ and $\beta$ parameters. The mode of the beta distribution is $\frac{\alpha - 1}{\alpha + \beta - 2}$, so picking an $\alpha$ and a $\beta$ that makes that fraction something like $0.1$ makes sense. How about $\alpha = 2, \beta = 10$?

That gives you a prior distribution that looks something like this:

The mode is right around 10%, the mean is around 17%. There is a small, but not negligible probability of a rate of around 50%. Or you could push more of the prior weight to values at or below 10% by using something like, $\alpha = 1, \beta = 9$:

It's easy enough to generate a beta distribution and visually inspect it. Here is some bare bones base R code:

alpha = 1
beta = 9
plot(x = seq(0.0001, 1, by = .0001), 
     y = dbeta(seq(0.0001, 1, by = .0001), alpha, beta),
     type = "l")

If you can make a plot match your prior intuition, then you have found your prior distribution.