Uniform (Noninformative) Prior and MLE

[Fei He]

Dear Stan Team,

You guys did great work! A question rises during my play with Stan. Hope you could help me with that.

Q1. In Page 6 of Stan-2.3.0 Reference, you mentioned that "If the prior is uniform, the posterior mode corresponds to the maximum likelihood estimate (MLE) of the parameters". If I want to get MLE of parameters in parameter block in Stan, in model block I should not give those parameters any prior, right? But when I tried comparing results between Stan and SAS (or NONMEM), the MLEs are quite different. Were the estimates by Stan with non-informative prior (priors not specified) really MLEs? I also saw examples of giving weak priors that make the R_hat much better. Do the results with weak priors differs a lot from MLEs?

Q2. If I want to put random effects in the model, then the results run by "optimize" is no longer the MLEs, right? Then I can only use "sample" and take the mean for random effects?

Your answers really help. Thank you so much!

Bests,
Fei

[Andrew Gelman]

Fei:

1. Stan gives mle (or, if there’s a prior, penalized mle) if you run it in optimize mode, but not if you run it in sampling mode.  Sampling mode is the default.  If you run stan in optimize mode with no piror and the point estimate is not a (local) maximum of the likelihood, there is a problem.

2.  Nonmem also gives marginal mle which is something different.  We’re planning to put marginal mle in Stan but it’s not there yet.

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