Page 286 - Modern Spatiotemporal Geostatistics

A Call to Research 267

development of powerful means of calculation—analytical and numerical (dia-
grammatic representations, perturbation expansions, Monte Carlo simulations,
etc.) The physical meaning of the mathematical terms, however, is not among
the issues that are discussed in this part. For example, from the viewpoint of
formal geostatistics, the shape of the general knowledge-based pdf is a mathe-
matical assumption that is made on the basis of internal consistency rules and
can be justified in terms of the predictive maps to which it leads.

Interpretive BME and the Search for "Rosebud"

Interpretive issues are relevant when we need to establish relationships between
the natural world in which we use the pdf of the physical maps, and the formal
mathematics which describe them, i.e., to measure and test formal structures
or to justify certain methodological steps of the mapping procedure. For ex-
ample, if explanation in terms of epistemic ideals is an issue for a study under
consideration, the principle of maximum expected information (or maximum
entropy; Chapter 5) can be used to justify the association of a particular pdf
with the physical map; other rules may be needed to translate mathemati-
cal expressions into testable statements, etc. Therefore, the interpretive part
examines carefully the physical content and scientific substance of the geosta-
tistical models.
Interpretation is an important component of applied stochastic analysis,
in general. While probability theory and statistics establish the mathematical
properties of stochastic concepts and tools, they do not tell us how to measure,
interpret, or derive them from physical data, laws, and theories. A physically
meaningful interpretation of probability, e.g., cannot be obtained by means of
statistical arguments, but rather by establishment of relationships between the
natural world in which we use probabilities and the stochastic mathematics
which describe them. This important fact which is at the heart of many phys-
ical scientists' criticism of statistical approaches has been acknowledged by a
number of statisticians, as well. Dempster and Wang, among others, have sug-
gested (see Wang, 1993, p. 87) that "although statistical model-building makes
use of formal probability calculations, the probabilities usually have no sharply
defined interpretation, so the whole model-building process is really a form of
exploratory analysis." One may also emphasize the fundamental differences—in
substance and scope—between statistical and scientific hypotheses (e.g., while
the former usually focus on a distinct feature of a specified population, the
latter involve a deeper understanding of the underlying physical mechanisms).
The interpretive part of BME involves both ontological and epistemic in-
vestigations. We will attempt a comparative discussion of these investigations
with the help of the masterpiece movie Citizen Kane. In this film, Kane's
famous last word was "rosebud." This word gave rise to an intriguing search
by reporters, movie historians, etc. regarding two central questions: "What
is 'rosebud'?" and "What does 'rosebud' mean?" The film's narrative thus
has two sides, each reflecting one of the two questions. While one concerns