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, introduced by Peter Green, which 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.

Software packages

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.

References

1. Green, P.J. (1995). "Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination". Biometrika. 82 (4): 711–732. CiteSeerX 10.1.1.407.8942. doi:10.1093/biomet/82.4.711. JSTOR 2337340. MR 1380810.