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Given the definition of exponential family $$ f(x\mid\theta) = \exp\left( \eta(\theta)^\top T(x) + h(x) - A(\theta) \right) $$ for $\theta\in\Theta$ and $x\in\mathcal{X}$, doesn't this property of sufficient statistic $$ T(x) = T(y), \forall x, y \in\mathcal{X} \Rightarrow f(x\mid\theta) = f(y\mid\theta), \forall \theta\in\Theta, \forall x, y \in\mathcal{X} $$ imply $h(x) = g(T(x)), \forall x\in\mathcal{X}$ for some $g$?

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