Theory of Statistics - George Mason University

Markov Chain Monte Carlo
4.7 Computational Methods 373
Monte Carlo techniques often allow us to make statistical inferences when the
statistical method involves intractable expressions. In applications in Bayesian
inference, we can study the posterior distribution, which may be intractable or
which may be known only proportionally, by studying random samples from
that distribution.
In parametric Bayesian inference, the objective is to obtain the conditional
posterior distribution of the parameter, given the observed data. This is QH
in equation (4.2), and it is defined by the density in step 5 in the procedure
outlined in Section 4.2.3. This density contains all of the information about
the parameter of interest, although we may wish to use it for specific types of
inference about the parameter, such as a point estimator or a credible set.
Understanding the Posterior Distribution
As with any probability distribution, a good way to understand the posterior
distribution is to take a random sample from it. In the case of the posterior
distribution, we cannot take a physical random sample. We can, however,
simulate a random sample, using methods discussed in Section 0.0.7, beginning
on page 657.
In single-parameter cases, random samples from the posterior distribution
can often be generated using a direct acceptance/rejection method if the
constant of proportionality is known. If the posterior density is known only
proportionally, a Metropolis-Hastings method often can be used.
Often the posterior density is a fairly complicated function, especially in
multi-parameter cases or in hierarchical models. In such cases, we may be able
to express the conditional density of each parameter given all of the other
parameters. In this case, it is fairly straightforward to use a Gibbs sampling
method to generate samples from the multivariate distribution. Consider the
relatively simple case in Example 4.5. The joint posterior PDF is given in
equation (4.32). We can get a better picture of this distribution by simulating
random observations from it. To do this we generate a realization σ 2 from
the marginal posterior with PDF given in equation (4.33), and then with that
value of σ 2 , we generate a realization µ from the conditional posterior with
PDF given in equation (4.34).
Example 4.19 illustrates this technique for a hierarchical model.
The simulated random samples from the posterior distribution gives us a
picture of the density. It is often useful to make pair-wise scatter plots of the
samples or estimated contour plots of the density based on the samples.
Simulated random samples can be used to approximate expectations of
functions of the random parameters with respect to the posterior density
(this is Monte Carlo quadrature), and they can also be used to identify other
properties of the posterior distribution, such as its mode.
Theory of Statistics c○2000–2013 James E. Gentle