Hong Jiao writes:

I work in the area of educational measurement and statistics. I am currently using the MCMC method to estimate model parameters for item response theory (IRT) models. Based on your book, Gelman and Hill (2007), the scale indeterminancy issue can be solved by constraining the mean of item parameters or the mean of person parameters to be 0. Some of my colleagues stated that by using the priors for either the item or person parameters, we do not need to set any constraints for the scale indeterminancy when using the MCMC for the IRT model estimation. They thought the use of the priors determine the scale. I am somewhat in disagreement with them but have nobody to confirm.

My response:

I recommend fitting the model with soft constraints (for example, by setting one of the prior means to 0 and imposing an inequality constraint somewhere to deal with the sign aliasing) and then using summarizing inferences using relative values of the parameters. The key idea is to post-process to get finite population inferences. We discuss this in one of the late chapters of our book and also in our Political Analysis article from 2005. (In particular, see Section 2 of that article.)

Constraining the prior mean is fine but it doesn’t really go far enough, I think. Ultimately it depends on what your inferential goals are, though.