# Natural parametrization of pdfs

1. Nov 21, 2011

### WantToBeSmart

I am struggling to understand the concept of natural parametrization of pdf of exponential family. Say that we have a function with the following pdf:

$$f(x;\theta)=exp\left[\sum_{j=1}^k A_j(\theta)B_j(x)+C(x)+D(\theta)\right]$$

where A and D are functions of $\theta$ alone and B and C are functions of x alone.

Natural parametrization.

$$f(x;\phi)=exp\left[\sum_{j=1}^k \phi_jB_j(x)+C(x)+D(\phi)\right]$$

where $$\phi_j=A_j(\theta)$$

My two questions are:

1 How to I find $D(\phi)?$
2 Can we perform natural parametrization on all pdfs belonging to the exponential family? If not why is that the case?

It isn't clear from your notation whether $\phi$ is a scalar or a vector.