Page 139 - Modern Spatiotemporal Geostatistics

120 Modern Spatiotemporal Geostatistics — Chapter 5
Specificatory Knowledge Base"). Approaches leading to different arrangements
of the knowledge bases incorporated in the prior and meta-prior stages can be
considered, a fact that adds to the flexibility of the BME analysis. Computa-
tional aspects can also play a role in decisions about the physical knowledge
to be used at each stage. In some cases, physical knowledge-based constraints
may arise for which pdf maximizing of the expected information (Eq. 5.2) is
difficult to obtain (see, e.g., Shimony, 1985). It is then possible that these
constraints can be put in a form that is easily incorporated at the meta-prior
stage.

The Integration or Posterior Stage

In the prior stage, the operator 9£ processed general knowledge Q. The Q
knowledge as well as the specificatory knowledge S considered in the meta-prior
stage are incorporated into the mapping process by means of a new operator
introduced by the following postulate.
POSTULATE 5.4: At the integration stage, the updated pdf of the map
should be expressed in terms of an operator % which integrates the
general knowledge-based operator 9£ with the specificatory knowledge
S considered at the meta-prior stage, i.e.,

where !?C = ^7 U 5, and A is a normalization parameter.

At this point, Equation 5.35 presents an abstract form of % which indi-
cates its dependence on the general knowledge operator %,, the specificatory
knowledge S, and the mapping vector x map = (Xhard. X soft, X k)\ the opera-
tor *y s is associated with the mapping point p k. Conceptually, Postulate 5.4
is a direct consequence of Proposition 4.1 (p. 95), in which case the operator
y s and the normalization parameter A account for the probability functions
/<? [Xmap(^)] and MXdata(>S)]> respectively. TheXfclXdataC^) stands for the
possible values Xk of the map in the context specified by Xdata(^) anc ' Q- Prob-
ability models such as Equation 5.35 should always be understood to apply in
the appropriate context, which defines the current state of physical knowledge.
Postulate 5.4 suggests a natural synthesis of the general knowledge of the
prior stage and the specificatory knowledge of the meta-prior stage. In the
multipoint case (see Chapter 4, p. 89), Postulate 5.4 is concerned with the
multivariate pdf f^(Xk)- Xk = (Xfcu- • • )XfcP)- m the single-point case, on the
other hand, we are dealing with several univariate pdf's f^(xki), i = 1, • • • ,p
(see Fig. 5.4).
Certain ^-operators have been introduced earlier in this chapter (p. 106).
In order, then, to obtain an explicit analytical form for y s, the form of the
specificatory knowledge S must first be described explicitly (in addition to the
hard facts, a geostatistician may find it useful to look to the judgments of those