Reversible-jump Markov chain Monte Carlo

In computational statistics, reversible-jump Markov chain Monte Carlo is an extension to standard Markov chain Monte Carlo (MCMC) methodology that allows simulation of the posterior distribution on spaces of varying dimensions.[1] Thus, the simulation is possible even if the number of parameters in the model is not known.

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.