Definition of exponential family with dispersion parameter

Question

I was recently reading a discussion of generalized linear models that considered the response to come from an exponential family with a dispersion parameter so $$ f(y|\theta,\phi) = \exp\left(\frac{y\theta - b(\theta)}{a(\phi)} + c(y, \phi)\right) $$ for known function $a$, $b$, and $c$. Everywhere else the official definition of an exponential family that I've seen is $$ g(y|\theta) = \exp(\theta^T T(y) - A(\theta))h(y). $$

How do I reconcile these two definitions? Is $f$ really something slightly different?

I could define a new parameter $\xi = \theta / a(\phi)$ and then I have $$ f(y) = \exp\left(y\xi - d(\theta,\phi)\right)c^*(y, \phi) $$ with $d(\theta,\phi) = b(\theta) / a(\phi)$ and $c^* = \exp \circ c$ but I've got $\phi$ potentially interacting with the support which isn't ok if it's supposed to be a parameter, and the partition function $d$ can depend on $\theta$ and $\phi$ separately rather than just on $\xi$. So that doesn't seem to make them coincide. What's going on here?

Answer 1

The definition you quote which is used with generalized linear models (glm) is not an exponential family, it is an exponential dispersion family. For a fixed value of the dispersion parameter $\phi$ it is an exponential family (indexed by $\theta$), but when $\phi$ varies it is not.

When used in glm's, the exponential dispersion family is used for inference about $\theta$, but eventual inference about the dispersion parameter $\phi$ is done outside that framework.