(This article was originally published at Statistical Modeling, Causal Inference, and Social Science, and syndicated at StatsBlogs.)
Luke Miratrix writes:
In the applied stats class I’m teaching on hierarchical models I’m giving the students (a mix of graduate students, many from the education school, and undergrads) a taste of Stan. I have to give them some “standard” way to turn Stan output into a point estimate (though of course I’ll also explain why credible intervals are better if they have the space to give them). But what I don’t know is how to tell them to do this? What is the state of the art?
First, let’s presume we’re talking about a parameter which is more or less well-behaved: clearly identifiable, reasonably unimodal, at least a few steps along the asymptotic path towards symmetry, etc. In that case, the question is mostly just an academic one, because anything we do will come up with more-or-less the same answer. In cases like that, should we use median, mean, trimmed mean, or what?
Second, what happens when things start to break down? We might be close to the edge of parameter space, or there might be some bimodal situation where two (or more) parameters can combine in two (or more) different plausible ways, and the evidence from the data isn’t yet enough to conclusively rule one of those out. In those cases, plotting the distribution should clearly show that something is up, but what do we do about it in terms of giving a point estimate? The best answer might be “don’t”, but sometimes it can be hard to tell that to the ref… In this kind of case, the mean might be the most defensible compromise, though of course it should come with caveats.
Basically, I need to give them some simple answer that is least likely to raise eyebrows when they’re trying to publish. Bonus points if it’s also a good answer!
My reply: If it’s one parameter at a time, I like the posterior median. Of course, in multiple dimensions the vector (median of parameter 1, median of parameter 2, . . ., median of parameter K) is not necessarily a good joint summary. So that’s a potential concern. No easy answers! There’s also some work on Bayesian point estimates. It turns out that the best prior, if you’re planning to compute a posterior mode, is not necessarily the prior you’d use for full Bayes:
http://www.stat.columbia.edu/~gelman/research/published/chung_etal_Pmetrika2013.pdf
http://www.stat.columbia.edu/~gelman/research/published/chung_cov_matrices.pdf
Bonus: these papers have education examples!
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