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The exponential family can be described by \begin{align*} f(y|\theta,\phi) = \exp\left\{\phi^{-1}(y\theta - b(\theta) + c(y,\phi)\right\} \end{align*}

It also can be shown that $\textbf{E}(Y) = \mu = b^{\prime}(\theta)$ and $\textbf{Var}(Y) = \phi\ b^{\prime\prime}(\theta)$. Based on this, we can define the variance function as $V(\mu) = b^{\prime\prime}(\theta)$. My question is: based on $V$, how do we recover $\theta$? The text which I am studying says that \begin{align*} \theta = \int V^{-1}(\mu)\mathrm{d}\mu \end{align*}

Although quite simple, I am having troubles in understanding this relation. Any help is appreciated. Thanks in advance!

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