Page 118 - Modern Spatiotemporal Geostatistics

The Epistemic Paradigm 99

is used in mathematics. Another sort of related conditional is the strong map
conditional, denoted by Xdata(S) ^ X*. which is, by definition, true if and
only if the Xdatatf) ~» X fc 's necessarily true.
In terms of the complementarity concept (discussed in Chapter 2, p. 59),
one can say that a material conditional holds between Xdata(^) anc ' Xk m
all possible random field realizations when Xdata(S) ar"d ""Xfc is not tne case-
In other words, material conditionally holds in every possible random field
realization except in those possible realizations in which the case is X dato(5)
and -* fc.
The conditional probability of a map is not necessarily the probability of
its material conditional. Whether these two are or are not equal may depend
on the knowledge base considered. In particular, the probability of truth of the
e
material map conditional "x dato(5) —> Xk >" i- ->

is not necessarily equal to Prob §[x fc|Xdoto(5)]. Prob ff[x fc|Xdota(^)]. in other
words, is the probability that \ k occurs given that Xdoto(^) does, whether or
not Xdata(S) S ave r se to Xk' an ^ whether or not there is any (probabilistic)
'
subjunctive connection between Xk anc ^ Xdata(<$)- '* can be shown that the
above map probabilities are related by

which implies that Prob§[x; fc|x data(5)] < Prob s[ Xdata(S) -> X k}- However,
in light of Equation 4.5 it is valid that

In other words, the meaning of Equation 4.20 is that, given that Xdata(S} has
indeed been observed, the probability of the material map conditional is equal
to the conditional probability of the map.
An interesting representation of the analysis above is obtained in the con-
text of map truth tables. It is instructive to illustrate this representation by
means of the following example.
EXAMPLE 4.6: Consider the map truth tables shown in Figure 4.2. Given
§, the Probg of the material map conditional Xdato(^) ~* Xk includes all
three realizations (x^aC^), X fe), (^X^S), xj, and (->X data(S), -xj of
the truth table in Figure 4.2a. On the other hand, the Abased Prob^ includes
only the realization (Xdota(^)> X/t) °f Figure 4.2b (i.e., from the probabilistic
logic viewpoint, if we select a world at random according to Prob^, the latter
gives the probability that we selected a world in which both Xdata(^) ^d
X fc occur). Note that nothing is written into the conditional map probability
XfclXdato(^) column when -x data(S) is considered (Fig. 4.2a).