Richard Zur writes,

Are any Bayesian estimates invariant to parameterization? If not, what do people do about it?

I was planning on constructing an informative prior in one parameterization and then reparameterizing into something more convenient. I was planning on using a MAP estimate to compare to the MLE, but now I’m worried because MAP is not invariant. What about mean, median, variance, etc? Do people deal with this issue anywhere? Would delving into the invariant prior literature help? Mostly they seem to focus on non-informative priors, as far as I can see.

My quick answer is that it’s ok for things to depend on parameterization; that is in fact a key way in which information is encoded in a model. Even linear transformations can affect how parameters are interpreted and how models are selected, thus affecting the final inferences. I’m not a big fan of invariant prior distributions (although we do discuss the topic briefly in Chapter 2 of Bayesian Data Analysis).

I’ll also use this to promote one of my favorite papers, Parameterization and Bayesian Modeling. Here’s the abstract:

Progress in statistical computation often leads to advances in statistical modeling. For example, it is surprisingly common that an existing model is reparameterized, solely for computational purposes, but then this new conŽ guration motivates a new family of models that is useful in applied statistics. One reason why this phenomenon may not have been noticed in statistics is that reparameterizations do not change the likelihood. In a Bayesian framework, however, a transformation of parameters typically suggests a new family of prior distributions. We discuss examples in censored and truncated data, mixture modeling, multivariate imputation, stochastic processes, and multilevel models.

and here’s the paper.