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##

Bayesian inference on uncertain
variables and posterior characterization

We start here with a few
one-dimensional problems involving
simple models that often occur in data analysis.
These examples will be used to illustrate some of the
most important Bayesian concepts.
Let us first introduce briefly the structure of the Bayes' theorem
in the form convenient to our purpose, as a straightforward
extension of what was seen in Sect. 4.2.

is the generic name of the parameter (used hereafter, unless
the models have traditional symbols for their parameters) and is the
data point.
is the prior,
the posterior
and
the likelihood.
Also in this case the
likelihood is is often written as
, and the same
words of caution expressed in Sect. 4.2 apply here too.
Note, moreover, that, while
is a properly
normalized pdf,
has not a pdf meaning in the
variable . Hence,
the integral of
over is only accidentally equal to unity.
The denominator in the r.h.s. of Eq. (24) is called the *evidence* and,
while in the parametric inference discussed here is just a trivial normalization factor,
its value becomes important for model comparison (see Sect. 7).
Posterior probability distributions
provide the full description of our state of knowledge about the
value of the quantity. In fact, they allow to calculate all
*probability intervals* of interest. Such intervals are
also called *credible intervals* (at a specified level of probability,
for example 95%)
or *confidence intervals* (at a specified level of 'confidence',
i.e. of probability).
However, the latter expression could be confused with
frequentistic 'confidence intervals', that are not probabilistic
statements about uncertain variables (D'Agostini 1999c).

It is often desirable to characterize the
distribution in terms of a few numbers. For example,
mean value (arithmetic *average*) of the posterior,
or its most probable value (the *mode*)
of the posterior, also known as the *maximum a
posteriori (MAP) estimate*.
The spread of the distribution is often
described in terms of its *standard deviation*
(square root of the *variance*).
It is useful to associate the terms mean value and standard deviation with
the more inferential terms *expected value*,
or simply *expectation* (value),
indicated by , and
*standard uncertainty* (ISO 1993), indicated by , where the
argument is the uncertain variable of interest.
This will be our standard way of reporting the result of
inference in a quantitative way, though, we emphasize
that the full answer is given by the posterior distribution,
and reporting only these summaries
in case of the complex distributions
(e.g. multimodal and/or asymmetrical pdf's)
can be misleading, because people tend to think of a
Gaussian model if no further information is provided.

** Next:** Gaussian model
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Giulio D'Agostini
2003-05-13