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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!

[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!

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