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##

Probability of complete classes

These formulae become more interesting when we consider a set of
propositions that all together form a tautology
(i.e., they are *exhaustive*) and are mutually *exclusive*.
Formally

When these conditions apply, the set
is said to form a *complete class*.
The symbol has been chosen because we shall soon interpret
as a set of *hypotheses*.
The first (trivial) property
of a complete class is *normalization*, that is

which is just an extension of Eq. (6)
to a complete class
containing more than just a single proposition and its negation.
For the complete class , the generalizations of
Eqs. (6) and the use of
Eq. (4) yield:

Equation (10) is the basis of what is called *marginalization*, which will become particularly important when
dealing with uncertain variables: the probability of is
obtained by the summation over all possible
*constituents*
contained in . Hereafter, we avoid explicitly writing the
limits of the summations, meaning that they extend over all
elements of the class. The constituents are `,' which,
based on the complete class of hypotheses , themselves form
a complete class, which can be easily proved.
Equation (11) shows that the probability of any proposition
is given by a weighted average of all conditional probabilities,
subject to hypotheses forming a complete class, with the weight
being the probability of the hypothesis.
In general, there are many ways to choose complete classes
(like `bases' in geometrical spaces). Let us denote the
elements of a second complete class by . The constituents are then
formed by the elements of the Cartesian product
. Equations (10) and
(11) then become the more general statements

and, symmetrically,

The reason we write these formulae both ways is to
stress the symmetry of Bayesian reasoning with respect to
classes and , though we shall soon associate them with *observations* (or *events*) and *hypotheses*, respectively.

** Next:** Probability rules for uncertain
** Up:** Rules of probability
** Previous:** Probability of simple propositions
Giulio D'Agostini
2003-05-13