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##

Multidimensional case -- Inferring and of a Gaussian

So far we have only inferred one parameter of a model.
The extension to many parameters
is straightforward. Calling
the set of parameters
and
the data, Bayes' theorem becomes

Equation (52) gives the posterior for the full
parameter vector
.
*Marginalization* (see Tab. 1)
allows one to
calculate the probability distribution for a single parameter,
for example,
,
by integrating over the remaining
parameters. The marginal distribution
is then the complete result of the Bayesian
inference on the parameter . Though the characterization
of the marginal is done in the usual way described in
Sect. 5.1, there is often the interest to
summarize some characters of the multi-dimensional
posterior that are unavoidably lost in the marginalization
(imagine marginalization as a kind of geometrical projection).
Useful quantities are the covariances between parameters
and , defined as

As is well know, quantities which give a more intuitive idea
of what is going on are the correlation coefficients, defined
as
.
Variances and covariances form the
covariance matrix
, with
and
.
We recall also that convenient formulae to calculate variances and covariances
are obtained from the expectation of the products
,
together with the expectations of the parameters:

As a first example of a multidimensional distribution from a
data set, we can think, again, at the inference of the parameter
of a Gaussian distribution, but in the case that also
is unknown and needs to be determined by the data.
From Eqs. (52), (50) and
(25), with and
and neglecting overall normalization, we obtain

The closed form of Eqs. (56) and 57)
depends on the prior and, perhaps, for
the most realistic choice of
, such a
compact solution does not exists. But this is not an essential issue,
given the present computational power.
(For example, the shape of
can be easily inspected
by a modern graphical tool.)
We want to stress here
the conceptual simplicity of the Bayesian solution to the problem.
[In the case the data set contains some more than a dozen of
observations, a flat
, with the constraint
, can be considered a good practical choice.]

** Next:** Predictive distributions
** Up:** Inferring numerical values of
** Previous:** Inference from a data
Giulio D'Agostini
2003-05-13