# 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

$n_m\in N_m=\{1,2,\ldots,I\} \,$

be a model indicator and $M=\bigcup_{n_m=1}^I \R^{d_m}$ the parameter space whose number of dimensions $d_m$ depends on the model $n_m$. The model indication need not be finite. The stationary distribution is the joint posterior distribution of $(M,N_m)$ that takes the values $(m,n_m)$.

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

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

The function

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

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

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

so that there exists an inverse function

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

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

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

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

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

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

with

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

The acceptance probability will be given by

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

where $|\cdot |$ denotes the absolute value and $p_mf_m$ is the joint posterior probability

$p_mf_m=c^{-1}p(y|m,n_m)p(m|n_m)p(n_m), \,$

where $c$ is the normalising constant.

## Software packages

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

## References

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