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 models; the model represents the key characteristics or behaviors of the selected system or process, whereas the ...
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 ...
on
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 con ...
s of varying
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coor ...
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 plans of a building in late 16th-century English, and derived via French and Italian ultimately from Latin ''modulus'', a measure.
Models c ...
is not known.
Let
:
be a model
indicator and
the parameter space whose number of dimensions
depends on the model
. 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
Traditionally, a finite verb (from la, fīnītus, past partici ...
. 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
:
The function
:
must be ''one to one'' and differentiable, and have a non-zero support:
:
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 ...
:
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
:
with
:
The acceptance probability will be given by
:
where
denotes the absolute value and
is the joint posterior probability
:
where
is the normalising constant.
Software packages
There is an experimental RJ-MCMC tool available for the open source
BUGs
Bugs may refer to:
* Plural of bug
Arts, entertainment and media Fictional characters
* Bugs Bunny, a character
* Bugs Meany, a character in the ''Encyclopedia Brown'' books
Films
* ''Bugs'' (2003 film), a science-fiction-horror film
* ''Bugs ...
package.
Th
Gen probabilistic programming systemautomates 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