If a distribution isn't in canonical form, can there be a natural parameter?

If a distribution isn't in canonical form, can there be a natural parameter?

• A. Yes
• B. No
• C. Only in some cases
• D. Sometimes

Answer: (B)

The natural parameter is defined within the exponential family when a distribution is written in its canonical form, f(x|η) = h(x) exp(η^T T(x) − A(η)). In this form, η is the natural parameter because the log-density is linear in the sufficient statistic T(x). If a distribution cannot be expressed in that canonical exponential-family form, there is no natural parameter for it in that representation. To have a natural parameter, you would need to reparameterize the model to bring it into exponential-family form; otherwise, in the given form, a natural parameter does not exist.

The natural parameter is defined within the exponential family when a distribution is written in its canonical form, f(x|η) = h(x) exp(η^T T(x) − A(η)). In this form, η is the natural parameter because the log-density is linear in the sufficient statistic T(x). If a distribution cannot be expressed in that canonical exponential-family form, there is no natural parameter for it in that representation. To have a natural parameter, you would need to reparameterize the model to bring it into exponential-family form; otherwise, in the given form, a natural parameter does not exist.