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Natural parametrization of pdfs 
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#1
Nov2111, 12:34 PM

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:
[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! 


#2
Nov2211, 01:44 PM

Sci Advisor
P: 3,321




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