The parameter that controls the heaviness of the tails in a t distribution is called the degrees-of-freedom parameter, ν (nu). I prefer to call it the normality parameter because larger values make the distribution more normal, and the distribution is normal when ν is infinity. Values of ν larger than about 30.0 make the distribution roughly normal, and values of ν less than about 30.0 give the distribution noticeably heavier tails than a normal.
Here's the news: Past Self had learned, sometime long ago, that ν must be 1.0 or larger as an algebraic constraint in the definition of the t distribution. Past Self was wrong. In fact, ν must be 0.0 or larger. This issue was brought to my attention by Osvaldo Martin who asked me, in comments on a previous blog post, why I thought ν had to be greater than 1.0. Perhaps other people have asked me this question in the past, but somehow this is the first time I was prompted to double check my answer.
So what? Well, all the models involving a t distribution in the 1st and 2nd editions of DBDA had priors that restricted ν to be 1.0 or greater (see figure below). Fortunately, there is essentially no bad consequence whatsoever, because real data virtually never are distributed with such extreme outliers that an estimated value of ν<1 is needed. This lack of consequence is presumably why the error of Past Self has gone uncorrected so long.
|The prior used in DBDA2E. There's no need to shift it +1.|
Despite the error being virtually harmless, I've changed all the programs so that the priors go from ν=0 up. This change simplifies the model specifications, too. Now the JAGS models simply specify
nu ~ dexp(1/30.0)
nu <- nuMinusOne + 1
nuMinusOne ~ dexp(1/29.0)
I've made this change in umpteen programs, including the Stan programs too. I've checked that all the JAGS programs still run fine, but I have not yet checked the Stan programs. Let me know if you encounter any problems. The programs are available at the book's web site.
The change also means that the diagrams of the models can be simplified, such as Figure 16.11 on p. 468. I'll leave that to your imagination.