# Natural parametrization of pdfs

by WantToBeSmart
Tags: natural, parametrization, pdfs
 P: 10 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!
 Quote by WantToBeSmart 1 How to I find $D(\phi)?$
It isn't clear from your notation whether $\phi$ is a scalar or a vector.