How should one go about turning a frequentist result into a Bayesian prior?
Consider the following pretty generic scenario: An experiment was conducted in the past and a result on some parameter $\phi$ was measured. The analysis was done with a frequentist methodology. A confidence interval for $\phi$ is given in the results.
I'm now conducting some new experiment where I want to measure some other parameters, say both $\theta$ and $\phi$. My experiment is different than the previous study --- it is not performed with the same methodology. I would like to do a Bayesian analysis, and so I will need to place priors on $\theta$ and $\phi$.
No previous measurements of $\theta$ have been performed, so I place a uninformative (say its uniform) prior on it.
As mentioned, there is a previous result for $\phi$, given as a confidence interval. To use that result in my current analysis, I would need to translate the previous frequentist result into an informative prior for my analysis.
One option that is unavailable in this made up scenario is to repeat the previous analysis that led to the $\phi$ measurement in a Bayesian fashion. If I could do this, $\phi$ would have a posterior from the previous experiment that I would then use as my prior, and there would be no issue.
How should I translate the frequentist CI into a Bayesian prior distribution for my analysis? Or in other words, how could I translate their frequentest result on $\phi$ into a posterior on $\phi$ that I would then use as a prior in my analysis?
Any insights or references that discuss this type of issue are welcome.