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:

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

where A and D are functions of [itex]\theta[/itex] alone and B and C are functions of x alone.

Natural parametrization.

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

where [tex]\phi_j=A_j(\theta)[/tex]

My two questions are:

1 How to I find [itex]D(\phi)?[/itex]
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

It isn't clear from your notation whether [itex] \phi [/itex] is a scalar or a vector.

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.