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 $p$.


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