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Suppose an acceptance ratio of the MH sampler without parameter transformation is like this: \begin{gather} r=\frac{f( \theta ',\gamma '|y,m) q( \theta ',\gamma '|\theta ,\gamma )}{f( \theta ,\gamma |y,m) q( \theta ,\gamma |\theta ',\gamma ')}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\frac{f( y|\theta ') f( m|\theta ',\gamma ') f( \theta ') f( \gamma ') q( \theta ',\gamma '|\theta ,\gamma )}{f( y|\theta ) f( m|\theta ,\gamma ) f( \theta ) f( \gamma ) q( \theta ,\gamma |\theta ',\gamma ')} , \notag \end{gather} where $\displaystyle y$ and $\displaystyle m$ are observed responses, $\displaystyle \theta $ and $\displaystyle \gamma $ are parameters $\displaystyle \in \mathbb{R}$ (the $\displaystyle '$ means proposal values), $\displaystyle q(\cdot|\cdot)$ is the proposal distribution.

However, in evaluating $\displaystyle f( m|\theta ',\gamma ')$ and $\displaystyle f( m|\theta ,\gamma )$, I actually need to firstly transform the $\displaystyle \theta $ and $\displaystyle \gamma $ (also $\displaystyle \theta '$ and $\displaystyle \gamma '$) by a logistic function to $\displaystyle t$ and $\displaystyle g$ whose support is restricted to [0, 1]. As a result, the acceptance ratio I really need to calculate is \begin{equation} r=\frac{f( y|\theta ') f( m|t',g') f( \theta ') f( \gamma ') q( \theta ',\gamma '|\theta ,\gamma )}{f( y|\theta ) f( m|t,g) f( \theta ) f( \gamma ) q( \theta ,\gamma |\theta ',\gamma ')} , \end{equation} So my question is do I need to add a jacobian term for this acceptance ratio?

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    $\begingroup$ Not at all since the reparameterisation here only impact the likelihood $f( y|\theta ) f( m|t,g)$ and neither the prior $f( \theta ) f( \gamma )$ nor the proposal. $\endgroup$
    – Xi'an
    Aug 9, 2020 at 14:21
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    $\begingroup$ @Xi'an Finally I can comment now... Thank you so much. Your answer helps me a lot! $\endgroup$
    – Keith Lau
    Jan 31, 2021 at 14:17

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