Let ${\displaystyle n_{m}\in N_{m}=\{1,2,\ldots ,I\}\,}$ be a model indicator and ${\displaystyle M=\bigcup _{n_{m}=1}^{I}\mathbb {R} ^{d_{m}}}$  the parameter space whose number of dimensions ${\displaystyle d_{m}}$  depends on the model ${\displaystyle n_{m}}$ . The model indication need not be finite. The stationary distribution is the joint posterior distribution of ${\displaystyle (M,N_{m})}$  that takes the values ${\displaystyle (m,n_{m})}$ .

The proposal ${\displaystyle m'}$  can be constructed with a mapping ${\displaystyle g_{1mm'}}$  of ${\displaystyle m}$  and ${\displaystyle u}$ , where ${\displaystyle u}$  is drawn from a random component ${\displaystyle U}$  with density ${\displaystyle q}$  on ${\displaystyle \mathbb {R} ^{d_{mm'}}}$ . The move to state ${\displaystyle (m',n_{m}')}$  can thus be formulated as

${\displaystyle (m',n_{m}')=(g_{1mm'}(m,u),n_{m}')\,}$ 

The function

${\displaystyle g_{mm'}:={\Bigg (}(m,u)\mapsto {\bigg (}(m',u')={\big (}g_{1mm'}(m,u),g_{2mm'}(m,u){\big )}{\bigg )}{\Bigg )}\,}$ 

must be one to one and differentiable, and have a non-zero support:

${\displaystyle \mathrm {supp} (g_{mm'})\neq \varnothing \,}$ 

so that there exists an inverse function

${\displaystyle g_{mm'}^{-1}=g_{m'm}\,}$ 

that is differentiable. Therefore, the ${\displaystyle (m,u)}$  and ${\displaystyle (m',u')}$  must be of equal dimension, which is the case if the dimension criterion

${\displaystyle d_{m}+d_{mm'}=d_{m'}+d_{m'm}\,}$ 

is met where ${\displaystyle d_{mm'}}$  is the dimension of ${\displaystyle u}$ . This is known as dimension matching.

If ${\displaystyle \mathbb {R} ^{d_{m}}\subset \mathbb {R} ^{d_{m'}}}$  then the dimensional matching condition can be reduced to

${\displaystyle d_{m}+d_{mm'}=d_{m'}\,}$ 

with

${\displaystyle (m,u)=g_{m'm}(m).\,}$ 

The acceptance probability will be given by

${\displaystyle a(m,m')=\min \left(1,{\frac {p_{m'm}p_{m'}f_{m'}(m')}{p_{mm'}q_{mm'}(m,u)p_{m}f_{m}(m)}}\left|\det \left({\frac {\partial g_{mm'}(m,u)}{\partial (m,u)}}\right)\right|\right),}$ 

where ${\displaystyle |\cdot |}$  denotes the absolute value and ${\displaystyle p_{m}f_{m}}$  is the joint posterior probability

${\displaystyle p_{m}f_{m}=c^{-1}p(y|m,n_{m})p(m|n_{m})p(n_{m}),\,}$ 

where ${\displaystyle c}$  is the normalising constant.

There is an experimental RJ-MCMC tool available for the open source BUGs package.

The Gen probabilistic programming system automates the acceptance probability computation for user-defined reversible jump MCMC kernels as part of its Involution MCMC feature.