Page 115 - Modern Spatiotemporal Geostatistics

96 Modern Spatiotemporal Geostatistics — Chapter 4

which expresses the obvious fact that given that the data vector Xdata(^)
has been observed—as the subscript ^ in Equation 4.5 indicates—its prob-
ability is one. On the other hand, at the prior stage it is usually valid that
Prob ff[x dota(5)] < 1. Furthermore, we have the following relation between
conditional probabilities

Equation 4.6 simply means that the evidential impact of X data(S} on Xk IS
already fully assessed in assigning the prior conditional probability and, hence,
the fact that Xdat a(<$) actually occurred is no reason to change this assessment.
[In other words, Equation 4.6 means that the probability we assign to \ k
assuming that Xdata(S) will turn out to be true is equal to the probability we
would assign to Xk on learning that Xdata(^) indeed turned out to be true.]
By the definition of conditional probability

which in light of Equation 4.5 gives

Taking Equation 4.6 into account, Equation 4.8 yields the Bayesian con-
ditionalization principle

which is the desired result. [This principle, in fact, was responsible for the
first letter in the acronym "BME," originally used to name the spatiotemporal
mapping approach; Christakos, 1990.] In light of Equation 4.9, the mathemat-
ical form of Prob^ is obtained from that of Prob $ by fixing Xdata(^) as tne
conditioning factor.

COMMENT 4.5 : Equation 4-4 ma y b e given th e following interpretation
in terms of the random field complementarity idea that was discussed in

Chapter 2 (p. 58): IfXdata(S) does indeed entail x k> thenx map(^} occurs in

s
every X-realization in which \ data(S) occurs. But ifXk ^ neither entailed

by Xdata(^} nor inconsistent with it, then Xmap(S) occurs only in some of

the possible X-realizations in which Xdata(S) occurs, ('i.e., those realizations

a
so

in which Xk ^ happens to occur). Therefore, we may take the ratio of

the quantity o f random fiel d realizations i n which Xmap(-5) occurs to th e

quantity o f realizations in which Xdota(^) occurs as determining th e extent

to which Xdata(S) entails Xk an< ^> thus, defining the probability ofx k given

Xdata^}} namely Prob^XfelXdataC^)]- **>ee a^so the discussion in the section

on conditional probability on p. 98.