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Reversible jump

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In recent years a method developed by Green ^[1] 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

$n_m\in N_m=\{1,2,\ldots,I\}$

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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 $g 1 m m'$ 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 1 m m' (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 m m' = d m' + d m' m

is met where $d m m'$ 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 m m' = d m'

with

(m, u) = g m' m (m)

The acceptance probability will be given by

$a(m,m')=min\left(1, \frac{p_{m'm}p_{m'}f_{m'}(m')}{p_{mm'}q_{mm'}(m,u)p_{m}f_m(m)}\left|det\left(\frac{\partial g_{mm'}(m,u)}{\partial (m,u)}\right)\right|\right),$