Reversible-jump Markov chain Monte Carlo

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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[edit]

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

References[edit]

  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.