I am a little confused about this. It was my understanding that exponential families are distinguished class of families of distributions.
For instance, they are regular (rather than singular).
The family of mixed Gaussians is not an exponential family I believe.
So my conclusion would be that the while “being Boltzmann” for a distribution is trivial as you point out, “being Boltzmann” (= exponential) for a family is nontrivial.
I am a little confused about this. It was my understanding that exponential families are distinguished class of families of distributions. For instance, they are regular (rather than singular).
The family of mixed Gaussians is not an exponential family I believe.
So my conclusion would be that the while “being Boltzmann” for a distribution is trivial as you point out, “being Boltzmann” (= exponential) for a family is nontrivial.