So far I have seen three formulations of the exponential family: $$ 1.\;\large f_Y(\mathbf{y}\,;\,\vec{\theta})=h(\mathbf{y}) \exp \left(\mathbf{b}(\vec{\theta}) \cdot \mathbf{T}(\mathbf{y})-A(\vec{\theta})\right) $$
$$ 2.\;\large f_Y(\mathbf{y}\,;\,\vec{\theta})=h(\mathbf{y}) \exp \left(\boldsymbol{\eta}(\vec{\theta}) \cdot \mathbf{T}(\mathbf{y})-A(\boldsymbol{\eta}(\vec{\theta}))\right) $$
$$ 3.\;\large f_Y(\mathbf{y}\,;\,\boldsymbol{\eta})=h(\mathbf{y}) \exp \left(\boldsymbol{\eta} \cdot \mathbf{T}(\mathbf{y})-A(\boldsymbol{\eta})\right) $$
- In formulation 1, $\vec{\theta}$ is called the canonical parameter whereas $\mathbf{b}(\mathbf{\vec \theta})$ is referred to as the natural parameter.
- In 2, $\mathbf{\vec \theta}$ is called both the canonical and natural parameter.
- In 3, $\boldsymbol{\eta}$ is called both the canonical and natural parameter.
To my understanding, an exponential family pdf is said to be in its canonical form when $\large \boldsymbol{\eta}(\mathbf{\vec \theta}) = \mathbf{\vec \theta}$ . So, in the canonical form, both the terms canonical parameter and natural parameter can be used interchangeably.
But what if the exponential family pdf is not in the canonical. Can they still be used interchangeably? I really need someone to clarify this terminology for me.