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In previous sections we have already seen some `tricks' for simplifying
the calculations. The main topic of this section will be
an introduction to Monte Carlo (MC). But, before doing that, we think
it is important to summarize the various `tricks' here.
Much specialized literature is available on several aspects of
computation in statistics.
For an excellent review paper on the subject see (Smith 1991).
- Conjugate priors
-
We discussed this topic in Sect. 8.3, giving a couple
of typical simple examples and references for a more detailed list of
famous conjugate distributions. We want to remark here that a conjugate prior
is a special case of the class of priors that simplify the calculation
of the posterior (the uniform prior is the simplest of this kind of prior).
- Gaussian approximation
-
For reasons that are connected with the central limit theorem,
when there is a large amount of consistent data the posterior
tends to be Gaussian, practically independently of the exact shape
of the prior. The (multi-variate)
Gaussian approximation, which we encountered in
Sect. 5.10, has an important role for applications,
either as a reasonable approximation of the `true' posterior, or as
a starting point for searching for a more accurate description of
it. We also saw that in the case of practically flat priors
this method recovers the well-known minimum chi-square or maximum
likelihood methods.
- Numerical integration
-
In the case of low dimensional problems, standard numerical integration
using either scientific library functions or the interactive tools of modern
computer packages provide an easy solution to many problems (thanks
also to the graphical capabilities of modern programs which allow
the shape of the posterior to be inspected and
the best calculation strategy decided upon).
This is a vast and growing subject, into
we cannot enter in any depth here, but we assume the
reader is familiar with some of these programs or packages.
- Monte Carlo methods
-
Monte Carlo methodology is a science in itself and it is
way beyond our remit
to provide an exhaustive introduction to it here.
Nevertheless, we would like to introduce briefly some 'modern'
(though the seminal work is already half a century old) methods
which are becoming extremely popular and are often associated
with Bayesian analysis, the so called
Markov Chain Monte Carlo (MCMC) methods.
Next: Monte Carlo methods
Up: Computational issues
Previous: Computational issues
Giulio D'Agostini
2003-05-13