# Making point estimates (with confidence intervals) part of a Stan model

Hi guys,

I would like to make point estimates (with confidence intervals) part of a bigger Bayesian model in the following sense: For one parameter M in my model I am given (\mu, \mu_l, \mu_u) corresponding to median, lower and upper bound of 95\% CI around the median. I do not have more information than this (except that the M is an estimated regression coefficient). The following two alternatives come to my mind on how to make use of M in my Bayesian (Stan) analysis:

1. Give M a normal prior with mean \mu and standard deviation that itself is given a weakly informative (hyper-)prior. Not sure what to do with \mu_l and \mu_u.
2. Model M with a normal likelihood with mean \tilde{\mu} and standard deviation \tilde{\sigma}, which in turn are given weakly informative priors. Now use the observed value \mu and condition on it. Not sure what to do with \mu_l and \mu_u.

Any ideas about the two approaches, recommendations about alternatives or literature (web) pointers?

From \mu_l and \mu_u you can deduce the standard error of \mu, since the CIs are roughly \mu\pm1.96*se, right? If the point estimates is an MLEs, then it is asymptotically normal distributed. So what about giving m a t prior like this M \sim t(\nu,\mu,se), where \nu reflects your â€śconfidenceâ€ť in these estimates?

I guess there is a more principled way of doing this, this is just what came to my mind. (So take with a grain of salt.)

Max

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Any suggestions from someone else?

If I had to pick something, I would go with the normal or t distribution as @Max_Mantei suggested. How was this regression coefficient computed? What was the sample size of the data set for that calculation? It might be perfectly reasonable to assume itâ€™s approximately normal since many estimators are asymptotically normal.

Iâ€™m approaching this as if youâ€™re not learning much about the parameter from your data (i.e., a sensitivity parameter). If you can learn about the parameter, it may not matter. If you canâ€™t, then I think you are leaving information on the table by not using \mu_l and \mu_u to inform prior uncertainty for M. I would prefer to be a little off about a distributional assumption than to wade into completely making up the uncertainty using weakly informative hyperpriors. With (\mu_l, \mu_u), you already have better than weak information, right?

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