It’s fine to use a vague prior for a Bayesian analysis when we really don’t have any idea about what a parameter is (or if we want to pretend we don’t), but when there is prior knowledge available, it’s often a waste to not draw from it. This prior knowledge might come from a non-statistician scientist, and so it’s important to have a way for them to communicate their knowledge to us in a way that allows us to construct a prior. However, unless the prior we want to construct is a Bernoulli or maybe a normal distribution, eliciting distribution parameters directly is unlikely to work.
Perhaps a more reliable and intuitive way is to specify quantiles, for example “we’re 50% sure that the proportion is less than 0.01 and 99% sure that it’s less than 0.05.” Giving two quantiles in this way is enough to determine a distribution from a two-parameter family, and is much easier to understand than saying that our prior knowledge follows a Beta(a, b) distribution.
Unless you’re much better at guessing parameters from quantiles than I am, it’s helpful to have a function that takes in the quantiles and returns the corresponding parameter values. Here’s the function I use for normal, beta, and gamma distributions, which are the three I most commonly use for priors (along with inverse-gamma priors, which can be obtained by inverting the desired quantiles in the gamma case).
The function is also available as a gist on Github.