# Terms and assumptions in trans-dimensional MCMC (RJ-MCMC) for Green 1995 paper

I want to use Trans-dimensional MCMC in my research and for fundamental understanding, I am trying to learn from Green (1995) paper, which is foundation of RJ-MCMC.

In part of 3.3 'switching between two simple subspace', it explains some mathematical foundation of proposed trans-dimensional MCMC. Some of the concepts and terms are not easy to understand.

In this paragraph, paper mention about "length" of $$u^1$$ and $$u^2$$ ($$m_1$$ and $$m_2$$) and also length of $$\theta^1$$ and $$\theta^2$$ $$(n_1,n_2)$$ must satisfy $$n_1+m_1=n_2+m_2$$. However, I am not certain I understand.

• What is the definition of "length" here?
• Aren't $$\theta^1$$ and $$\theta^2$$ parameter of current and proposed ones in birth/death step, and $$u^1$$ and $$u^2$$ are some perturbation here?
• Also not sure about deriving acceptance ratio of trans-dimensional step

I guess $$A$$ and $$B$$ is the subspace of $$k$$-dimension and $$k'$$-dimension and the equations derive balance between subspaces.

However, I am not get it what "Lebesgue measure" wants to achieve here and what is the meaning of second equations. Also, for me,it is unclear where Jacobian comes out in this equation.

It would be helpful if you can give any suggestion of study materials (set theory or some mathematical analysis) to understand the underlying concepts of trans-dimensional MCMC.

The key idea here is that we want a ratio of forward and backward proposal measures that is finite and non-zero. If $$x$$ and $$x'$$ have different numbers of dimensions, you will get infinite or zero ratios, and a reversible chain isn't possible. In particular, in the very common case where we specify measures on the two subspaces by densities wrt Lebesgue measure, we need a common base measure for the two densities we are dividing, which will also be a Lebesgue measure.
The idea in RJMCMC is that an auxiliary variable supplies the extra dimensions, so that the spaces $$(x,u_1)$$ and $$(x',u_2)$$ are the same dimension. If $$x$$ is four-dimensional and $$x'$$ is five-dimensional, and you need a two-dimensional additional random variable to propose $$x'\mapsto x$$, you will need a three-dimensional additional random variable to propose $$x\mapsto x'$$, so that we are mapping to and from $$\mathbb{R}^8$$.
An important point is that we don't need to keep all those auxiliary variables around (in contrast to some previous methods). At any given proposal we only need need as many auxiliary dimensions as the difference between the current $$x$$ and $$x'$$, and these are supplied by $$u_1$$ and $$u_2$$, which we would need anyway for generating the proposals.