Estimation in Financial Models - RiskLab

f() with respect to (jx). If 1 ; 2 ;:::; t ;:::is a realization from a suitable
chain then
t ,!
in distribution as t tends to innity, (jx) and
1
t
tX
i=1
f( i ) ,! E jx [f()] a.s.
as t tends to innity. Nowwe need algorithms to construct such chains with
specied equilibrium distributions. We will discuss two particular forms of
Markov chain schemes, the Gibbs sampling algorithm and the Metropolis{
Hastings algorithm.
The Gibbs sampling algorithm
Suppose = ( 1 ;:::; k ) is the vector of unknown quantities. We want to
simulate ( i j x) for i = 1;:::;k. Denote by (j x) = ( 1 ;:::; k j x) the
joint density and by ( i j x; j ;j 6= i) the so called induced full conditional
densities for each of the components i given values of the other components
j , j 6= i, for i =1;:::;k. Suppose that we are able to sample from each of
these one-dimensional distributions.
The Gibbs sampling algorithm (see Geman and Geman [29]):
1) choose arbitrary starting points 0 =( 0 1;:::; 0 k).
2) make random drawings from the full conditional distribution as follows
1 1
from ( 1 jx; 0 j ;j 6= 1)
1 2
from ( 2 jx; 1 1; 0 3;:::; 0 k)
1 3
from ( 3 jx; 1 1; 1 2; 0 4;:::; 0 k)
:::
1 k from ( k jx; i j;j 6= k)
This completes a transition from 0 =( 0 1;:::; 0 k) to 1 =( 1 1;:::; 1 k).
3) iterating step 2) produces a sequence 0 ; 1 ;:::; t ;::: which is a realization
of a Markov chain with transition probability K from t to
t+1 kY
K( t ; t+1 )=
l=1
t+1
l
x; t j;j >l; t+1
j ;j