My watch list
my.chemeurope.com  
Login  

Reversible jump



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\}

be a model indicator and M=\bigcup_{n_m=1}^I \R^{d_m} the parameter space whose number of dimensions dm depends on the model nm. The model indication need not be finite. The stationary distribution is the joint posterior distribution of (M,Nm) that takes the values (m,nm).

The proposal m' can be constructed with a mapping g1mm' 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',nm') can thus be formulated as

(m',nm') = (nm',g1mm'(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

dm + dmm' = dm' + dm'm

is met where dmm' 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

dm + dmm' = dm'

with

(m,u) = gm'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 pmfm is the joint posterior probability

pmfm = c − 1p(y | m,nm)p(m | nm)p(nm),

where c is the normalising constant.

  1. ^ P. J. Green. Reversible jump markov chain monte carlo computation and bayesian model determination. Biometrika, 82(4):711–732, 1995
 
This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Reversible_jump". A list of authors is available in Wikipedia.
Your browser is not current. Microsoft Internet Explorer 6.0 does not support some functions on Chemie.DE