0
$\begingroup$

Suppose you have a random variable X, who's distribution depends on $\theta$. If X is a part of the exponential family of distributions, X can be written in a certain form, namely: $$f_\theta(x)=h(x)*c(\theta)*e^{T(x)*\zeta(\theta)}$$

Is the above expression the 'Canonical' form of the distribution? If not, what is the relationship between the Canonical form of a distribution which is part of the exponential family (or any distribution if I am mistaken).

$\endgroup$
0
$\begingroup$

In the canonical parameterisation of an exponential family, the parameters appear as such in the scalar product, rather than being transformed. A canonical representation of a general exponential family is thus associated with a family of densities of the form $$f(x;\theta) = h(x) \exp\{ \theta^\text{T}T(x)-\Psi(\theta)\}$$ But this is not a unique representation as

  • full rank linear transforms of $T$, $T'(x)=A T(x)$, lead to $\theta'=A^{-1}\theta$
  • without the minimality constraint, components of $\theta$ and $T$ may be linearly dependent, while some components of $\theta$ may be fixed.
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.