Reversible-jump Markov chain Monte Carlo explained

In computational statistics, reversible-jump Markov chain Monte Carlo is an extension to standard Markov chain Monte Carlo (MCMC) methodology, introduced by Peter Green, which allows simulation (the creation of samples) 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. The "jump" refers to the switching from one parameter space to another during the running of the chain. RJMCMC is useful to compare models of different dimension to see which one fits the data best. It is also useful for predictions of new data points, because we do not need to choose and fix a model, RJMCMC can directly predict the new values for all the models at the same time. Models that suit the data best will be chosen more frequently then the poorer ones.

Details on the RJMCMC process

Let

nm\inNm=\{1,2,\ldots,I\}

be a model indicator and
I
M=cup
nm=1
dm
\R
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
dmm'
\R
. The move to state

(m',nm')

can thus be formulated as

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

The function

gmm':=((m,u)\mapsto((m',u')=(g1mm'(m,u),g2mm'(m,u))))

must be one to one and differentiable, and have a non-zero support:

supp(gmm')\ne\varnothing

-1
g
mm'

=gm'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

dm
\R

\subset

dm'
\R
then the dimensional matchingcondition 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,

pm'mpm'fm'(m')\left|\det\left(
pmm'qmm'(m,u)pmfm(m)
\partialgmm'(m,u)
\partial(m,u)

\right)\right|\right),

where

||

denotes the absolute value and

pmfm

is the joint posterior probability

pmf

-1
m=c

p(y|m,nm)p(m|nm)p(nm),

where

c

is the normalising constant.

Software packages

There is an experimental RJ-MCMC tool available for the open source BUGs package.

The Gen probabilistic programming system automates the acceptance probability computation for user-defined reversible jump MCMC kernels as part of its Involution MCMC feature.

References

  1. Green . P.J. . Peter Green (statistician) . 1995 . Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination . . 82 . 4 . 711–732 . 10.1093/biomet/82.4.711 . 1380810 . 2337340 . 10.1.1.407.8942 .