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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 [[space]]s of varying [[dimension]]s.<ref>{{cite journal
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| | last = Green |first= P.J. |authorlink=Peter Green (statistician)
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| | year = 1995
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| | title = Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination
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| | journal = [[Biometrika]]
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| | volume = 82
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| | issue = 4
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| | pages = 711–732
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| | doi = 10.1093/biomet/82.4.711
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| |mr=1380810 | jstor = 2337340
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| }}</ref>
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| Thus, the simulation is possible even if the number of [[parameter]]s in the [[Mathematical model|model]] is not known.
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| Let
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| :<math>n_m\in N_m=\{1,2,\ldots,I\} \, </math>
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| be a model [[indicator variable|indicator]] and <math>M=\bigcup_{n_m=1}^I \R^{d_m}</math> the parameter space whose number of dimensions <math>d_m</math> depends on the model <math>n_m</math>. The model indication need not be [[Wikt:finite|finite]]. The stationary distribution is the joint posterior distribution of <math>(M,N_m)</math> that takes the values <math>(m,n_m)</math>.
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| The proposal <math>m'</math> can be constructed with a [[map (mathematics)|mapping]] <math>g_{1mm'}</math> of <math>m</math> and <math>u</math>, where <math>u</math> is drawn from a random component
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| <math>U</math> with density <math>q</math> on <math>\R^{d_{mm'}}</math>. The move to state <math>(m',n_m')</math> can thus be formulated as
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| :<math>
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| (m',n_m')=(g_{1mm'}(m,u),n_m') \,
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| </math>
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| The function
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| :<math>
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| g_{mm'}:=\Bigg((m,u)\mapsto \bigg((m',u')=\big(g_{1mm'}(m,u),g_{2mm'}(m,u)\big)\bigg)\Bigg) \,
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| </math>
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| must be ''one to one'' and differentiable, and have a non-zero support:
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| :<math> \mathrm{supp}(g_{mm'})\ne \varnothing \, </math>
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| so that there exists an [[inverse function]]
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| :<math>g^{-1}_{mm'}=g_{m'm} \, </math>
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| that is differentiable. Therefore, the <math>(m,u)</math> and <math>(m',u')</math> must be of equal dimension, which is the case if the dimension criterion
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| :<math>d_m+d_{mm'}=d_{m'}+d_{m'm} \, </math> | |
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| is met where <math>d_{mm'}</math> is the dimension of <math>u</math>. This is known as ''dimension matching''.
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| If <math>\R^{d_m}\subset \R^{d_{m'}}</math> then the dimensional matching
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| condition can be reduced to
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| :<math>d_m+d_{mm'}=d_{m'} \, </math>
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| with
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| :<math>(m,u)=g_{m'm}(m). \, </math>
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| The acceptance probability will be given by
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| :<math>
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| a(m,m')=\min\left(1,
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| \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),
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| </math>
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| where <math>|\cdot |</math> denotes the absolute value and <math>p_mf_m</math> is the joint posterior probability
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| :<math>
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| p_mf_m=c^{-1}p(y|m,n_m)p(m|n_m)p(n_m), \,
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| </math>
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| where <math>c</math> is the normalising constant.
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| == Software packages ==
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| There is an experimental RJ-MCMC tool available for the open source [[BUGS]] package.
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| ==References==
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| <references/>
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| [[Category:Computational statistics]]
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| [[Category:Monte Carlo methods]]
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