# Jacobian term for Metropolis Hastings algorithm?

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 $$$$r=\frac{f( y|\theta ') f( m|t',g') f( \theta ') f( \gamma ') q( \theta ',\gamma '|\theta ,\gamma )}{f( y|\theta ) f( m|t,g) f( \theta ) f( \gamma ) q( \theta ,\gamma |\theta ',\gamma ')} ,$$$$ So my question is do I need to add a jacobian term for this acceptance ratio?

• 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. Aug 9, 2020 at 14:21
• @Xi'an Finally I can comment now... Thank you so much. Your answer helps me a lot! Jan 31, 2021 at 14:17