Natural parametrization of pdfs

In summary, the concept of natural parametrization involves rewriting a function with a multivariate pdf as a sum of functions involving only one variable. This can be done for some functions, but for others it may be more complex and there is no general procedure for finding the corresponding function in the new parametrization.
  • #1
WantToBeSmart
10
0
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!
 
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  • #2
WantToBeSmart said:
1 How to I find [itex]D(\phi)?[/itex]
It isn't clear from your notation whether [itex] \phi [/itex] is a scalar or a vector.

2 Can we perform natural parametrization on all pdfs belonging to the exponential family? If not why is that the case?

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.
 

1. What is a natural parametrization of pdfs?

A natural parametrization of pdfs refers to a way of representing the probability distribution function (pdf) of a random variable in a form that is most suitable for the particular variable and its characteristics. It involves choosing parameters that have a direct physical interpretation and make intuitive sense in the context of the problem at hand.

2. How is a natural parametrization different from other parametrization methods?

Natural parametrization differs from other methods in that it aims to choose parameters that are meaningful and have a direct relation to the variable being studied, rather than just being convenient mathematical constructs. This can help in better understanding and interpreting the results of statistical analyses.

3. What are some common examples of natural parametrization of pdfs?

Some common examples of natural parametrization of pdfs include using the mean and standard deviation to represent a normal distribution, using the shape and scale parameters to represent a gamma distribution, and using the location and scale parameters to represent a Weibull distribution.

4. How does natural parametrization impact statistical analyses?

Natural parametrization can greatly impact statistical analyses by making the results more interpretable and intuitive. It can also help in choosing appropriate models and making more accurate predictions based on the data. Additionally, natural parametrization can lead to more efficient and stable algorithms for numerical computations.

5. Are there any limitations to using natural parametrization of pdfs?

While natural parametrization has many advantages, it may not always be possible or practical to use in all cases. Some distributions do not have natural or meaningful parameters, and in those cases, other parametrization methods may be more suitable. Additionally, choosing the right natural parameters can sometimes be subjective and require expert knowledge.

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