Normal-Gamma: Metropolis-Hastings on log-scale, but no Jacobian?

I am reading the paper by Griffin and Brown (2010) where at one step in their MCMC procedure they need to sample from the following conditional posterior:

$$p(\lambda|\gamma, \Psi)\propto \pi(\lambda)\frac{1}{(2\gamma^2)^{p\lambda}(\Gamma(\lambda))^p}\left(\prod_{i=1}^p\Psi_i\right)^\lambda$$

They say that $$\lambda$$

can be updated using a Metropolis-Hastings random walk update on $$\log\lambda$$. We propose $$λ' = \exp\{\sigma^2_\lambda z\}\lambda$$, where $$z$$ is standard normal then $$\lambda'$$ is accepted with probability $$\min\left\{1, \frac{\pi(\lambda')}{\pi(\lambda)}\left(\frac{\Gamma(\lambda)}{\Gamma(\lambda')}\right)^p\left((2\gamma^2)^{-p}\prod_{i=1}^p\Psi_i\right)^{\lambda'-\lambda}\right\}$$

My question: why is there no term in the acceptance probability that accounts for the fact that we are proposing new draws on the logarithmic scale?

See for example here: Sampling on a logarithmic scale The paper, however, has more than 300 citations and I know several successful papers that have used the same type of Metropolis-Hastings procedure; therefore, I'm inclined to think I am missing something. But what?

Griffin and Brown (2010) Inference with normal-gamma prior distributions in regression problems, Bayesian Analysis, https://projecteuclid.org/euclid.ba/1340369797

• I would also deem the Jacobian is indeed missing from the Metropolis-Hastings ratio. – Xi'an Jun 12 at 20:18
• @Xi'an Thanks Xi'an, if you say so I feel more confident including the Jacobian part in my own implementation. Some limited experiments I have made show that it doesn't make a big difference, the posterior distribution of the first-level parameters ($\Psi$ here) is roughly the same. I guess that might be why they didn't notice they were using the wrong ratio (and why others haven't done so either when building onto their work). – jacknick Jun 13 at 6:17
• It would not make much of a difference if the posterior is sufficiently concentrated. – Xi'an Jun 13 at 8:00