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

Modeling
1

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?

2

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 “confidence” 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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3

Any suggestions from someone else?

4

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 it’s approximately normal since many estimators are asymptotically normal.

I’m approaching this as if you’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 can’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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