View source for Reversible-jump Markov chain Monte Carlo

{{short description|Simulation method in statistics}}
In computational statistics, '''reversible-jump Markov chain Monte Carlo''' is an extension to standard [[Markov chain Monte Carlo]] (MCMC) methodology, introduced by [[Peter Green (statistician)|Peter Green]], which allows [[simulation]] (the creation of [[Sampling (statistics)|samples]]) of the [[posterior distribution]] on [[space]]s of varying [[dimension]]s.<ref>{{cite journal
 | last = Green |first= P.J. |authorlink=Peter Green (statistician)
 | year = 1995
 | title = Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination
 | journal = [[Biometrika]]
 | volume = 82
 | issue = 4
 | pages = 711–732
 | doi = 10.1093/biomet/82.4.711
 |mr=1380810 | jstor = 2337340 
|citeseerx= 10.1.1.407.8942 }}</ref>
Thus, the simulation is possible even if the number of [[parameter]]s in the [[Mathematical model|model]] is not known. The "jump" refers to the switching from one parameter space to another during the running of the chain. '''RJMCMC''' is useful to compare models of different dimension to see which one fits the data best. It is also useful for predictions of new data points, because we do not need to choose and fix a model, '''RJMCMC''' can directly predict the new values for all the models at the same time. Models that suit the data best will be chosen more frequently than the poorer ones. 

== Details on the RJMCMC process ==
Let <math>n_m\in N_m=\{1,2,\ldots,I\} \, </math>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>.

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
<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

:<math>
  (m',n_m')=(g_{1mm'}(m,u),n_m') \, 
</math>

The function

:<math>
  g_{mm'}:=\Bigg((m,u)\mapsto \bigg((m',u')=\big(g_{1mm'}(m,u),g_{2mm'}(m,u)\big)\bigg)\Bigg) \, 
</math>

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

:<math> \mathrm{supp}(g_{mm'})\ne \varnothing \, </math>

so that there exists an [[inverse function]] 

:<math>g^{-1}_{mm'}=g_{m'm} \, </math>

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 

:<math>d_m+d_{mm'}=d_{m'}+d_{m'm} \, </math>

is met where <math>d_{mm'}</math> is the dimension of <math>u</math>. This is known as ''dimension matching''. 

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

:<math>d_m+d_{mm'}=d_{m'} \, </math>

with

:<math>(m,u)=g_{m'm}(m). \, </math>

The acceptance probability will be given by

:<math>
  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),
</math>

where <math>|\cdot |</math> denotes the absolute value and <math>p_mf_m</math> is the joint posterior probability

:<math>
  p_mf_m=c^{-1}p(y|m,n_m)p(m|n_m)p(n_m), \, 
</math>

where <math>c</math> is the normalising constant.

== Software packages ==
There is an experimental RJ-MCMC tool available for the open source [[BUGs (statistics)|BUGs]] package.

The [http://www.gen.dev Gen probabilistic programming system] automates the acceptance probability computation for user-defined reversible jump MCMC kernels as part of its [https://www.gen.dev/dev/ref/mcmc/#Involution-MCMC-1 Involution MCMC feature].

==References==
<references/>

[[Category:Computational statistics]]
[[Category:Markov chain Monte Carlo]]