Reversible-jump Markov chain Monte Carlo
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. Thus, the simulation is possible even if the number of parameters in the model is not known.
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
must be one to one and differentiable, and have a non-zero support:
so that there exists an inverse function
that is differentiable. Therefore, the and must be of equal dimension, which is the case if the dimension criterion
is met where is the dimension of . This is known as dimension matching.
If then the dimensional matching condition can be reduced to
The acceptance probability will be given by
where denotes the absolute value and is the joint posterior probability
where is the normalising constant.
There is an experimental RJ-MCMC tool available for the open source BUGS package.