A simulation method in statistics
In computational statistics, reversible-jump Markov chain Monte Carlo is an extension to standard Markov chain Monte Carlo (MCMC) methodology that allows simulation of the posterior distribution on spaces of varying dimensions.[1]
Thus, the simulation is possible even if the number of parameters in the model is not known.
Let
![{\displaystyle n_{m}\in N_{m}=\{1,2,\ldots ,I\}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ddffa2cb8a8b14bdd4a11546cdc42961d468825)
be a model indicator and
the parameter space whose number of dimensions
depends on the model
. The model indication need not be finite. The stationary distribution is the joint posterior distribution of
that takes the values
.
The proposal
can be constructed with a mapping
of
and
, where
is drawn from a random component
with density
on
. The move to state
can thus be formulated as
![{\displaystyle (m',n_{m}')=(g_{1mm'}(m,u),n_{m}')\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0fa49632dacc9efb8beb4a9d013ad73f954b75e)
The function
![{\displaystyle g_{mm'}:={\Bigg (}(m,u)\mapsto {\bigg (}(m',u')={\big (}g_{1mm'}(m,u),g_{2mm'}(m,u){\big )}{\bigg )}{\Bigg )}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9a3a06b77a0cdd80c760e067cb397d0c3f5a747)
must be one to one and differentiable, and have a non-zero support:
![{\displaystyle \mathrm {supp} (g_{mm'})\neq \varnothing \,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4269a57e5ae92e7c2d066c6253344f2e15ceb0)
so that there exists an inverse function
![{\displaystyle g_{mm'}^{-1}=g_{m'm}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f03378898c66fd21b9d3e622093567fa7da4843f)
that is differentiable. Therefore, the
and
must be of equal dimension, which is the case if the dimension criterion
![{\displaystyle d_{m}+d_{mm'}=d_{m'}+d_{m'm}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ed55ef5c480cfff34449b7ee296624ab0d0b1bf9)
is met where
is the dimension of
. This is known as dimension matching.
If
then the dimensional matching
condition can be reduced to
![{\displaystyle d_{m}+d_{mm'}=d_{m'}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e571b7bf7322f7f09a35f9227f35ff9ed2d0b9f)
with
![{\displaystyle (m,u)=g_{m'm}(m).\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c633888da701623ee1bcd5ab5d0d7b0e1a8f2f6c)
The acceptance probability will be given by
![{\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),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a0230646bfbab97aa64480d08d87ef8ddb30a07)
where
denotes the absolute value and
is the joint posterior probability
![{\displaystyle p_{m}f_{m}=c^{-1}p(y|m,n_{m})p(m|n_{m})p(n_{m}),\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b1733ebae7cd2b13e484bd7aefa774b8d2d363c)
where
is the normalising constant.
Software packages
There is an experimental RJ-MCMC tool available for the open source BUGs package.
References