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I read that given a parameter $\theta$ and a transformation $\phi = g(\theta)$ (where $\theta$ is the parameter of your prior distribution), the distribution of the transformed parameter would be:

$p_{\phi}(\phi) = p_{\theta}(g^{-1}(\phi))|\frac{d\theta}{d\phi}| = p_{\theta}(\theta)|\frac{d\theta}{d\phi}|$

Can someone explain to me how the second part of the RHS equation occurs, ie $|\frac{d\theta}{d\phi}|$ ?

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This normalization is important to ensure that the new pdf integrates to one. See for instance https://en.wikipedia.org/wiki/Random_variable, or the much better treatment given in Mood, Graybill and Boes (1974, chapter 5) which comprises a lot of details on functions of random variables.

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