# Canonical form and exponential family

Suppose you have a random variable X, who's distribution depends on $$\theta$$. If X is a part of the exponential family of distributions, X can be written in a certain form, namely: $$f_\theta(x)=h(x)*c(\theta)*e^{T(x)*\zeta(\theta)}$$

Is the above expression the 'Canonical' form of the distribution? If not, what is the relationship between the Canonical form of a distribution which is part of the exponential family (or any distribution if I am mistaken).

In the canonical parameterisation of an exponential family, the parameters appear as such in the scalar product, rather than being transformed. A canonical representation of a general exponential family is thus associated with a family of densities of the form $$f(x;\theta) = h(x) \exp\{ \theta^\text{T}T(x)-\Psi(\theta)\}$$ But this is not a unique representation as
• full rank linear transforms of $$T$$, $$T'(x)=A T(x)$$, lead to $$\theta'=A^{-1}\theta$$
• without the minimality constraint, components of $$\theta$$ and $$T$$ may be linearly dependent, while some components of $$\theta$$ may be fixed.