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Natural parametrization of pdfs

  1. Nov 21, 2011 #1
    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!
     
    Last edited: Nov 21, 2011
  2. jcsd
  3. Nov 22, 2011 #2

    Stephen Tashi

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    Science Advisor

    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.
     
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