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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
| | Emilia Shryock is my title but you can call me something you like. My working day job is a meter reader. Years ago we moved to North Dakota. One of the things she enjoys most is to do aerobics and now she is trying to earn cash with it.<br><br>My weblog ... std testing at home ([http://www.buzzbit.net/blog/288042 a knockout post]) |
| | 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> | |
| 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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Emilia Shryock is my title but you can call me something you like. My working day job is a meter reader. Years ago we moved to North Dakota. One of the things she enjoys most is to do aerobics and now she is trying to earn cash with it.
My weblog ... std testing at home (a knockout post)