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 $M=\bigcup _{{n_{m}=1}}^{I}\mathbb{R} ^{{d_{m}}}$ 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 $\mathbb{R} ^{{d_{{mm'}}}}$ . 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'}})\neq \varnothing \,

so that there exists an inverse function

g_{{mm'}}^{{-1}}=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 $\mathbb{R} ^{{d_{m}}}\subset \mathbb{R} ^{{d_{{m'}}}}$ 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 $p_{m}f_{m}$ is the joint posterior probability

p_{m}f_{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

↑ 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.

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