Next: Metropolis-Hastings Algorithm
Up: Simulation Methods Based on
Previous: Example: Hard-Core Model
  Contents
Gibbs Sampler
The MCMC algorithm for the generation of ,,randomly picked''
admissible configurations of the hard core model (see
Section 3.3.1) is a special case of a so-called Gibbs sampler for the simulation of discrete (high dimensional)
random vectors.
- MCMC Simulation Algorithm
-
Theorem 3.11
Let the transition matrix
be given
as
|
(36) |
where the conditional probabilities
are defined in
. Then
is
irreducible and aperiodic and the pair
is
reversible.
- Proof
-
The assertion can be proved similarly to the proof of
Theorem 3.10.
- In order to see that
is aperiodic it
suffices to notice
- that for all
- and hence all diagonal elements
of
are
positive.
- The following considerations show that
is irreducible.
- It is left to show that the detailed balance equation
(2.85) holds, i.e.
|
(38) |
Let
be a Markov chain with state space
and the transition matrix
given by (36). As a consequence of
Theorem 3.11 we get that in this case
|
(39) |
for any initial concentration
where
denotes the distribution of
. Furthermore, the Gibbs
sampler shows the following monotonic behavior.
- Proof
-
- Remarks
-
Theorem 3.13
The matrix
has the following
representation
|
(45) |
i.e., the multiplicative reversible version
of the
,,forward-scan matrix''
coincides with the
,,forward-backward scan matrix''.
- Proof
-
- Remarks
-
- If Gibbs samplers are used in practice it is always assumed
- The family
of subsets of is called
a system of neighborhoods if for arbitrary
- (a)
-
,
- (b)
-
implies
.
- For the hard-core model from Section 3.3.1,
is the set of those vertices that are
directly connected to by an edge.
Next: Metropolis-Hastings Algorithm
Up: Simulation Methods Based on
Previous: Example: Hard-Core Model
  Contents
Ursa Pantle
2006-07-20