# Reagarding the base measure h(x) in the exponential family

most authors define the canonical form of the exponential family as

$$p(\mathbf{x} | \boldsymbol{\theta})=h(\mathbf{x}) \exp (\boldsymbol{\eta}(\boldsymbol{\theta}) \cdot \mathbf{T}(\mathbf{x})-A(\boldsymbol{\theta}))$$

with the restriction that $$h(x)$$ must be non-negative. Why does one not simply define instead

$$p(\mathbf{x} | \boldsymbol{\theta})= \exp (g(\mathbf{x}) + \boldsymbol{\eta}(\boldsymbol{\theta}) \cdot \mathbf{T}(\mathbf{x})-A(\boldsymbol{\theta}))$$

with $$g(\mathbf{x}) = \log h(\mathbf x)$$, allowing us to drop the non-negativity assumption since it is sufficed automatically. This also lets us interpret $$g(\mathbf{x})$$ as a bias term added to the inner product $$\langle\boldsymbol{\eta}(\boldsymbol{\theta}), \mathbf{T}(\mathbf{x})\rangle$$

## 1 Answer

The first form makes more sense if you think about there being continuous and discrete exponential families. Then $$h$$ is a known probability measure that is absolutely continuous with respect to either Lebesgue or counting measure and the $$\exp(\eta(\theta) T(x) - A(\theta))$$ part is the density (or Radon–Nikodym derivative) with respect to this probability measure $$h$$.

In this interpretation the first representation is intuitive since we usually write probability measures that are indexed by densities as base measure times density.