I am studying some properties of exponential family distributions, i.e., distributions whose pdf/pmf can be written (in its "natural" form) as $$f_X(\mathbf{x}\mid\boldsymbol \theta) = h(\mathbf{x}) \exp\Big(\boldsymbol\eta({\boldsymbol \theta}) \cdot \mathbf{T}(\mathbf{x}) - A({\boldsymbol \theta})\Big). $$

In this Wikipedia page I found that $$E[T_j] = \frac{\partial A(\eta)}{\partial \eta_j}$$ where $T_j$ is the $j$th sufficient statistic and $\eta_j$ is the $j$th natural paramter.

If I view the expected value of sufficient statistic as a function of the natural paramters, i.e., $\mu({\boldsymbol\eta}) = E[\mathbf{T}(x)|{\boldsymbol\eta}]$, is this function one-to-one? In particular, given the following identify $$E[\mathbf{T}(x)|{\boldsymbol\eta_1}] = E[\mathbf{T}(x)|{\boldsymbol\eta_2}]$$ can we conclude that ${\boldsymbol\eta_1} = {\boldsymbol\eta_2}$?

I try reading different texts and online materials but could not find a conclusion. But I have a hard time constructing a counterexample.

It would be great if anyone can provide any insight into proving the above conclusion or constructing any counterexample. Thanks in advance!


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