In computational statistics, reversible-jump Markov chain Monte Carlo is an extension to standard

Markov chain Monte Carlo In statistics, Markov chain Monte Carlo (MCMC) methods comprise a class of algorithms for sampling from a probability distribution. By constructing a Markov chain that has the desired distribution as its equilibrium distribution, one can obtain ...

(MCMC) methodology, introduced by Peter Green, which allows

simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...

of the

posterior distribution The posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior p ...

space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually consider ...

s of varying

dimension In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...

s. Thus, the simulation is possible even if the number of

parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

s in the

model A model is an informative representation of an object, person or system. The term originally denoted the Plan_(drawing), plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a mea ...

is not known. Let :

n_m\in N_m=\ \,

be a model

indicator Indicator may refer to: Biology * Environmental indicator of environmental health (pressures, conditions and responses) * Ecological indicator of ecosystem health (ecological processes) * Health indicator, which is used to describe the health o ...

and

M=\bigcup_^I \R^

the parameter space whose number of dimensions

d_m

depends on the model

n_m

. The model indication need not be

finite Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

. 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_

m

and

u

, where

u

is drawn from a random component

U

with density

q

\R^

. The move to state

(m',n_m')

can thus be formulated as :

(m',n_m')=(g_(m,u),n_m') \,

The function :

g_:=\Bigg((m,u)\mapsto \bigg((m',u')=\big(g_(m,u),g_(m,u)\big)\bigg)\Bigg) \,

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

\mathrm(g_)\ne \varnothing \,

so that there exists an

inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\t ...

g^_=g_ \,

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_=d_+d_ \,

is met where

d_

is the dimension of

u

. This is known as ''dimension matching''. If

\R^\subset \R^

then the dimensional matching condition can be reduced to :

d_m+d_=d_ \,

with :

(m,u)=g_(m). \,

The acceptance probability will be given by :

a(m,m')=\min\left(1,
  \frac\left, \det\left(\frac\right)\\right),

where

, \cdot ,

denotes the absolute value and

p_mf_m

is the joint posterior probability :

p_mf_m=c^{-1}p(y, m,n_m)p(m, n_m)p(n_m), \,

where

c

is the normalising constant.

Software packages

There is an experimental RJ-MCMC tool available for the open source BUGs package. Th
Gen probabilistic programming system
automates the acceptance probability computation for user-defined reversible jump MCMC kernels as part of it
Involution MCMC feature

References

Computational statistics Markov chain Monte Carlo