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If I understand correctly, a distribution in the exponential family... $$\underline X\sim f_{\underline\theta}(\underline x) = exp\{\sum\limits_{i}\eta_i(\underline\theta)T_i(\underline x)-B(\underline\theta)\}~h(\underline x)$$ ...where $\underline\eta(\centerdot)$ is a (possibly vector valued) parameter, $\underline T(\centerdot)$ is a corresponding vector of sufficient statistics, $B(\theta)$ is the log partition, and $h(\underline x)$ is the base measure and $\underline T(\underline x)$ itself has the following distribution:

$$\underline T(\underline x) =\underline t\sim g_{\underline\theta}(\underline t) = exp\{\sum\limits_{i}\eta_i(\underline\theta)t_i-B(\underline\theta)\}~h^*(\underline t)$$

...where $h^*(\centerdot)$ is not necessarily the same function as the one from $f_{\underline\theta}(\underline x)$ above.

My questions are:

  • Are the other two functions ($B(\underline\theta)$ and $\underline\eta(\underline\theta)$) identical to the ones from $f_{\underline\theta}(\underline x)$?
  • What is the general strategy for finding $h^*(\centerdot)$?
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The functions are identical (same carrier measure, same log-normalizer).

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