0
$\begingroup$

In probability theory, a parameterized distribution belongs to the exponential family if it can be written in the form $$ p(\mathbf{x}; \boldsymbol \theta) = h(\mathbf{x}) \exp\Big(\boldsymbol\eta({\boldsymbol \theta}) \cdot \mathbf{T}(\mathbf{x}) - A({\boldsymbol \theta})\Big).$$

Given an arbitrary function $ p(\mathbf{x}; \boldsymbol \theta)$, is there a simple criterion to determine if there exist suitable functions $\boldsymbol\eta({\boldsymbol \theta})$ and $A({\boldsymbol \theta})$ to make this work?

I suspect the answer might lie in information geometry, but would prefer something simpler, for example a partial differential equation on $p$.

$\endgroup$

Your Answer

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

Browse other questions tagged or ask your own question.