Reversible jump

In recent years a method developed by Green allows simulation of the posterior distribution on spaces of varying dimensions. 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')=(n_m',g_{1mm'}(m,u)) $$

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, differentiable, and have a non-zero support


 * $$ \sup(gmm')\ne \varnothing $$

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.