Theory of Statistics - George Mason University

664 0 Statistical Mathematics
Generate x (k+1)
d−1 conditionally on x (k+1)
1 , x (k+1)
2 , . . ., x (k)
d ,
Generate x (k+1)
d conditionally on x (k+1)
1 , x (k+1)
2 , . . ., x (k+1)
d−1 .
3. If convergence has occurred, then
3.a. deliver x = x (k+1) ;
otherwise
3.b. set k = k + 1, and go to step 2.
Example 0.0.10 Gibbs Sampling to Generate Independent Normals
Consider Xt+1 normal with a mean of Xt and a variance of σ 2 .
We will generate an iid sample from a standard normal distribution; that
is, a normal with a mean of 0 and a variance of 1. In this example, the target
distribution is simpler than the proposal.
We start with a x0, chosen arbitrarily.
We take logs and cancel terms in the expression for r.
The following simple Matlab statements generate the sample.
x(1) = x0;
while i < n
i = i + 1;
yip1 = yi + sigma*randn;
lr2 = yi^2 - yip1^2;
if lr2 > 0
yi = yip1;
else
u = rand;
if lr2 > log(u)*2
yi = yip1;
else
yi = x(i-1);
end
end
x(i) = yi;
end
plot (x)
There are several variations of the basic Metropolis-Hastings algorithm.
Two common related methods are Gibbs sampling and hit-and-run sampling.
Those methods are particularly useful in multivariate simulation.
Markov chain Monte Carlo has become one of the most important tools in
statistics in recent years. Its applications pervade Bayesian analysis, as well
as many Monte Carlo procedures in the frequentist approach to statistical
analysis.
Whenever a correlated sequence such as a Markov chain is used, variance
estimation must be performed with some care. In the more common cases
Theory of Statistics c○2000–2013 James E. Gentle