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Reversible-jump Markov chain Monte Carlo
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<h2 id="details-on-the-rjmcmc-process">Details on the RJMCMC process</h2> <p>Let n m ∈ N m = { 1 , 2 , … , I } {\displaystyle n_{m}\in N_{m}=\{1,2,\ldots ,I\}\,} be a model <a href="/facts/Indicator_variable/LbT5Wcjm">indicator</a> and M = ⋃ n m = 1 I R d m {\displaystyle M=\bigcup _{n_{m}=1}^{I}\mathbb {R} ^{d_{m}}} the parameter space whose number of dimensions d m {\displaystyle d_{m}} depends on the model n m {\displaystyle n_{m}} . The model indication need not be finite. The stationary distribution is the joint posterior distribution of ( M , N m ) {\displaystyle (M,N_{m})} that takes the values ( m , n m ) {\displaystyle (m,n_{m})} . </p><p>The proposal m ′ {\displaystyle m'} can be constructed with a <a href="/facts/Map_(mathematics)/FABF1l2f">mapping</a> g 1 m m ′ {\displaystyle g_{1mm'}} of m {\displaystyle m} and u {\displaystyle u} , where u {\displaystyle u} is drawn from a random component U {\displaystyle U} with density q {\displaystyle q} on R d m m ′ {\displaystyle \mathbb {R} ^{d_{mm'}}} . The move to state ( m ′ , n m ′ ) {\displaystyle (m',n_{m}')} can thus be formulated as </p> ( m ′ , n m ′ ) = ( g 1 m m ′ ( m , u ) , n m ′ ) {\displaystyle (m',n_{m}')=(g_{1mm'}(m,u),n_{m}')\,} <p>The function </p> g m m ′ := ( ( m , u ) ↦ ( ( m ′ , u ′ ) = ( g 1 m m ′ ( m , u ) , g 2 m m ′ ( m , u ) ) ) ) {\displaystyle g_{mm'}:={\Bigg (}(m,u)\mapsto {\bigg (}(m',u')={\big (}g_{1mm'}(m,u),g_{2mm'}(m,u){\big )}{\bigg )}{\Bigg )}\,} <p>must be <i>one to one</i> and differentiable, and have a non-zero support: </p> s u p p ( g m m ′ ) ≠ ∅ {\displaystyle \mathrm {supp} (g_{mm'})\neq \varnothing \,} <p>so that there exists an <a href="/facts/Inverse_function/Tg5nnXlu">inverse function</a> </p> g m m ′ − 1 = g m ′ m {\displaystyle g_{mm'}^{-1}=g_{m'm}\,} <p>that is differentiable. Therefore, the ( m , u ) {\displaystyle (m,u)} and ( m ′ , u ′ ) {\displaystyle (m',u')} must be of equal dimension, which is the case if the dimension criterion </p> d m + d m m ′ = d m ′ + d m ′ m {\displaystyle d_{m}+d_{mm'}=d_{m'}+d_{m'm}\,} <p>is met where d m m ′ {\displaystyle d_{mm'}} is the dimension of u {\displaystyle u} . This is known as <i>dimension matching</i>. </p><p>If R d m ⊂ R d m ′ {\displaystyle \mathbb {R} ^{d_{m}}\subset \mathbb {R} ^{d_{m'}}} then the dimensional matching condition can be reduced to </p> d m + d m m ′ = d m ′ {\displaystyle d_{m}+d_{mm'}=d_{m'}\,} <p>with </p> ( m , u ) = g m ′ m ( m ) . {\displaystyle (m,u)=g_{m'm}(m).\,} <p>The acceptance probability will be given by </p> a ( m , m ′ ) = min ( 1 , p m ′ m p m ′ f m ′ ( m ′ ) p m m ′ q m m ′ ( m , u ) p m f m ( m ) | det ( ∂ g m m ′ ( m , u ) ∂ ( m , u ) ) | ) , {\displaystyle 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),} <p>where | ⋅ | {\displaystyle |\cdot |} denotes the absolute value and p m f m {\displaystyle p_{m}f_{m}} is the joint posterior probability </p> p m f m = c − 1 p ( y | m , n m ) p ( m | n m ) p ( n m ) , {\displaystyle p_{m}f_{m}=c^{-1}p(y|m,n_{m})p(m|n_{m})p(n_{m}),\,} <p>where c {\displaystyle c} is the normalising constant. </p> <h2 id="software-packages">Software packages</h2> <p>There is an experimental RJ-MCMC tool available for the open source <a href="/facts/BUGs_(statistics)/W8vpNEy4">BUGs</a> package. </p><p>The <a href="https://www.gen.dev">Gen probabilistic programming system</a> automates the acceptance probability computation for user-defined reversible jump MCMC kernels as part of its <a href="https://www.gen.dev/dev/ref/mcmc/#Involution-MCMC-1">Involution MCMC feature</a>. </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Green, P.J. (1995). "Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination". Biometrika. 82 (4): 711–732. CiteSeerX 10.1.1.407.8942. doi:10.1093/biomet/82.4.711. JSTOR 2337340. MR 1380810. <a href="/wiki/Peter_Green_(statistician)" target="_blank">/wiki/Peter_Green_(statistician)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> </ol>