# Name and interpretation of "$h(x)$" in exponential family

The exponential family is defined (in many sources) as:

$$p(x | \theta) = h(x) \exp\{\theta^TT(x) - A(\theta)\}$$

where:

• $$T(x)$$ is a sufficient statistic,
• $$\theta$$ is a canonical parameter, and
• $$A(\theta)$$ is a cumulant function

What is $$h(x)$$? Does it have a name or specific interpretation?

The $$h(x)$$ function in the exponential family is known as the "underlying measure." It serves to ensure $$x$$ is in the right space. For many functions, this correction is unnecessary (i.e. it is set to $$1$$ or $$1/\sqrt{2}$$). It does play a strong role in defining many functions, however. Since the role is function-specific beyond the definition above, I will link to a part of the Wikipedia page for "exponential family" with a few helpful examples (in table form) of the role of $$h(x)$$ in common distributions.
• (+1) I would add that $h(\cdot)$ is the determinantal element of the family. Once the reference measure is chosen, plus the sufficient statistic $T(\cdot)$, the parameter space is naturally defined as made of these $\theta$'s for which the density is well-defined and $A(\theta)$ is the logarithm of the normalising constant. Commented Jun 5, 2019 at 18:49