# Trying to understand Reversible Jump MCMC

As stated in the foundational Biometrika paper of Green (1995) 'Reversible Jump Monte Carlo Calculation and model discrimination'

I am researching the inversion of the data in Geophysics and MCMC is suitable for what I am doing. Furthermore, the model I try to use can have a different number of parameters, which is also needed in trans-dimensional MCMC often called "Reversible-Jump MCMC"

However, I have trouble understanding concepts and implementation in my case. From what I understand from the paper by Green, 1995 and some other literature, the acceptance ratio of RJ-MCMC can be calculated from the following equations Min(1, (Likelihood Ratio)(prior ratio)(Proposal Ratio)*(Jacobian)) For the birth/death step in MCMC, the concept of ''Proposal ratio' elude me to some extent. For my understanding, it is intended to balance between different dimensions. At Green (1995), the proposal ratio is calculated as $$d_{k+1}L/(b_k(k+1))$$, where $$L$$ is the length of time in the simulation and $$k$$ is the current 'change point' so $$(k+1)$$ is the dimension test for the acceptance.

The aspect that confuses me here is "L" representing the time duration, where the change point can locate, and the potential infinities of change point positioning between 0 and "L". Because, for the birth step, a point can be suggested in any location between 0 and "L", which effectively create an infinite range. For the "death" step, the selection of the removing point is among (k+1), therefore proposal probability of the "death" step is $$1/(k+1)$$ and the proposal probability of the "birth" step is $$1/L$$. And L becomes infinity if we can suggest any position between 0 and $$L$$. Then the "proposal ratio" can intend to balance between birth-death $$L \text{(infinity)} /(k+1)$$, which I feel something wrong here.

What am I missing here? If this process is for reversibility, I guess the proposal ratio of birth/death should be 1 if I "propose" birth/death in an equal ratio. However, the reference regarding RJMCMC states otherwise. Or does RJMCMC need some fixed "grid" where the change point should be located between "0" and "L"?

• Could you point to the source of the equation you provided for the proposal ratio? The ratio of the proposals should describe how likely it is to transition from the current parameter configuration to the proposed configuration and vice versa. From what I understand, it's the Jacobian that would handle the difference in dimensions. Commented Aug 12, 2023 at 14:11
• @fm361 Thanks for your comment! I hope I don't care much about 'proposal ratio' The source of equation is from paper of Green (1995) 'Reversible Jump Monte Carlo Calculation and model discrimination' academic.oup.com/biomet/article-abstract/82/4/711/252058 I also add part of paper regarding 'proposal ratio' Commented Aug 12, 2023 at 17:33
• Note that "L becomes infinity if we can suggest any position between 0 and L" makes no sense. $L$ remains $L$; the proposal is uniformly distributed over $(0, L)$ and consequently the proposal density is $1/L$. Commented Aug 12, 2023 at 17:49
• @jbowman Thanks.I am getting confusion here. Then How to determine unit of "L" when calculating 'proposal ratio'? What if parameter is within time interval and new "birth" is generated from time interval (0,L)? If new "birth" can be suggested in any range of "L" not from 1,2,3,4,.....,L, then proposal ratio become 1/L not make sense to me. Commented Aug 12, 2023 at 18:11
• Do you know what a continuous probability distribution is? Specifically, the Uniform distribution? (If not, I suggest that RJMCMC is not an appropriate thing to study at this point in your career...) en.wikipedia.org/wiki/Continuous_uniform_distribution Commented Aug 12, 2023 at 18:14