# Exponential family admissibility of base measure, sufficient statistic and log partition function

Let $$f(y | \eta) = h(y) \exp\left( \eta^\top T(y) + A(\eta) \right)$$ be the exponential family with base density/pmf $$h$$, sufficient statistic $$T$$, log partition function $$A$$ and natural parameter $$\eta$$. If it helps, we may restrict it to the case where $$\eta$$ is scalar. Obviously $$h$$ must be nonnegative. One can show that $$A$$ must also be convex.

The function $$A(\eta)$$ ... is automatically determined once the other functions have been chosen, since it must assume a form that causes the distribution to be normalized (sum or integrate to one over the entire domain).

Bold s on functions by me for emphasis.

My Question:

Given some fixed $$T$$, am I free to choose any convex $$A$$ such that there exists an $$h$$ such that the resulting function is a valid pdf/pmf? More precisely, given some $$T$$ consider the set $$\mathcal{A}_T = \{ A:\mathbb{R} \to \mathbb{R} \mid \exists \, h:\mathbb{R} \to \mathbb{R} \mid A(\eta) = \log\int h(y)\exp \left(\eta T(y) \right)\, dy\}.$$ Is there a useful alternative characterisation of this set?

As rightly pointed by the Wikipedia page, $$A(\cdot)$$ is completely determined by the choice of the triplet
dominating measure $$\text dy$$ x base density $$h(\cdot)$$ x statistic $$T(\cdot)$$
Up to a constant, $$A(\cdot)$$ is a Laplace transform and hence satisfies all properties of Laplace transforms, including identifying the associated distribution (of $$T$$).
Choosing first $$A(\cdot)$$ out of the blue and then looking for an associated triplet does not seem like a fruitful endeavour.
• I appreciate your response, even though my question is evidently not clear. I understand that $A(\cdot)$ is must satisfy $A(\eta) = \log \int h(y) \exp\left( \eta^\top T(y) \right) \, d y$ and can see how it can be used to identify the cumulants of $T$. The reason I would like to choose $A$ first is that I find myself in the situation where I am forced to use a specific $A$ and am wondering whether the associated subset of exponential families is useful. Is there anything I can add to my question to make it better? Jul 15, 2021 at 8:11
• A useful item of information would be to explain why you choose $A(\cdot)$ first. Otherwise, I do not for one see how one can answer the question. Jul 15, 2021 at 15:56