Natural parametrization of pdfs

[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?

Thank you in advance!

 

[Stephen Tashi]

1 How to I find $D(\phi)?$

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

2 Can we perform natural parametrization on all pdfs belonging to the exponential family? If not why is that the case?

I had to look up "natural parameterization" on the web and from that glance, I think what you are asking entails such questions as: How do we write an arbitrary function of two variables as a sum of two functions, each involving one variable? I don't know a procedure for doing that. If you allowed to define new variables then for f(x,y) = xy, one could certainly set p = xy and write f(x,y) as a function of one variable p. For a more complicated function, I think it is a complicated question. Perhaps some other forum member will know the answer.