Page 111 - Modern Spatiotemporal Geostatistics

92 Modern Spatiotemporal Geostatistics — Chapter 4
A simple yet illuminating (and, perhaps, entertaining) example may help
fix certain ideas about the aforementioned epistemic stages.
EXAMPLE 4.1: Imagine Nature as a furnished house. At the prior stage our
general knowledge Q is that Nature bought the furniture from KMart (but we
have not seen any specific articles). On the basis of Q, we may derive some
conclusions about the total value of the furniture (e.g., no article is worth
more than $200). At the meta-prior stage we inspect certain articles and we
obtain their actual values (case-specific knowledge S), which improves our prior
estimate of the total value of the furniture.
Epistemically important qualities of the mapping paradigm are those that
can be determined before the event (i.e., before the phenomenon predicted by
the map actually takes place). With this in mind, we want maps that carry
as much information as possible, that are well-supported with evidence, and
that have a high probability of being correct (rather than certain correctness,
which is an aspect that cannot be guaranteed before the event).
Reflecting upon such an epistemic paradigm, the BME mapping approach
was proposed about a decade ago (Christakos, 1990, 1991a, 1992). The crux
of BME is that the spatiotemporal analysis and mapping of natural phenom-
ena should be both informative and cogent. Due to the natural variations
and uncertainties involved in the description of such phenomena, both of these
requirements involve probabilities, but they are conditional probabilities rela-
tive to the different knowledge bases considered at each stage. This double
epistemic goal of BME is summarized by the following postulate.
POSTULATE 4.1: BME aims at informativeness (in terms of prior in-
formation relative to the general knowledge (?) as well as cogency (in
terms of posterior probability relative to the specificatory knowledge S).
The two epistemic ideals of Postulate 4.1 are at the conceptual heart of BME.
Let us, therefore, examine them in more detail. [Multipoint mapping will be
considered in this section; the single-point mapping can always be derived as
a special case.]

Prior stage

At the prior stage, the probability function considered is relative to the general
knowledge Q, i.e.,

which means "the probability of the mapx map = (x^, Xk) S iven tne general
knowledge base Q is p." Another way of expressing the meaning of Equa-
tion 4.1 is by saying that probability judgments about \ map are relative to
knowledge Q. Furthermore, an interpretation can be given in terms of the
complementarity idea of Chapter 2 (p. 58): Given §, the probability function
(Eq. 4.1) provides a measure of the relative quantity of random field realizations
in which Xma P occurs over all possible realizations.