Hi all,

I have some survey data where respondents are asked to guess a median and 90% interval for the probability of X. So for example, a respondent might say:

- 5th quantile: .001
- Median: .01
- 95th quantile: .1

I’m hoping to fit a beta distribution to these data in a way that uses the full uncertainty given by the respondents.

### My approach doesn’t work

My first thought was to use each respondent’s quantiles and ask ‘what is the Beta distribution that has these quantiles?’, then use that as the basis for a measurement error/missing data approach to accounting for the uncertainty. But this doesn’t always work, especially when the quantiles are all in one extreme – I can’t always get a Beta distribution with the given quantiles.

### This other approach seems like a great alternative

Recently I was pointed towards Ben Goodrich’s 2020 StanCon talk where he talks about the advantages of using an Inverse CDF to define priors using elicited quantiles. This seems to be the paper that the above StanCon talk became? Ben also seems to have implemented some of these distributions in Stan at the time.

### My question:

Has anyone been working with Quantile-Parameterized Distributions in Stan in the years since Ben Goodrich’s talk? I’m thinking of using some of the code from the above repo for my analysis, but I’m not sure which make the most sense for my research context. It would also be great to see more examples of Stan programs that use these as priors, if more examples exist.

Thanks in advance!