Page 143 - Modern Spatiotemporal Geostatistics

124 Modern Spatiotemporal Geostatistics — Chapter 5

usually much simpler than what most geostatisticians are capable of handling.
In fact, the most important problems are conceptual rather than mathematical.

COMMENT 5.8: A s already mentioned, th e characterization "Bayesian" i n

the acronym BME denotes the fact that Bayesian conditionalization is used

in th e theoretical analysis of the integration stage of the approach (see, e.g.,

Proposition 4-1 of Chapter 4, P- 95). In the eyes of some statisticians, this

analysis may not in the orthodox Bayesian framework. This is hardly

surprising. As Wang (1993; p.158) notices: "There are at least 46,656 vari-
eties of Bayesians." This statement is, obviously, a hyperbole that serves to

stress the fact that there are, indeed, various kinds of Bayesian approaches,

including the orthodox, the subjective, and the epistemic Bayesianisms. On

the other hand, the characterization "Maximum Entropy" in the acronym

BME is due to the fact that Equation 5.2 has the mathematical form of the
entropy function used by Boltzmann and Jaynes (in thermodynamics and

statistical mechanics; e.g., Boltzmann, 1964 [1896-98], Jaynes, 1983), b y
Shannon (in the description, storage, and transmission of messages; Shan-

non, 1948), an d b y many others (see, e.g., Ebanks et al. , 1998). I n al l

the above cases, however, the entropy functions were developed in different

scientific contexts than the spatiotemporal mapping situation considered in

Equation 5.2. As a matter of fact, it is not uncommon in scientific inves-
tigations to start from different origins and to end up with similar math-

ematical formulations of otherwise different physical situations. Entropy
is a case in point. In 1896, the term was introduced by Boltzmann in the

kinetic theory of gases in an effort to measure disorder by means of the
probabilities of molecular arrangements. In 1948, while working on com-

munication engineering problems, Shannon derived a working definition of

syntactic information which, when translated into mathematical symbols,

was identical to the Boltzmann entropy function. Certainly, from a physi-

cal interpretation point of view, the two entropies were drastically different.

In light of these examples, we conclude that the mathematical form of en-
tropy arises in various scientific disciplines, in which, however, it has very

different physical interpretations.

By way of a summary, the problem of how to integrate the ^-base (scien-
tific theories, physical laws, etc.) with the 5-base (case-specific data, empirical
evidence, etc.) when constructing a space/time map has generated consider-
able controversy in geostatistics and spatial statistics. The BME approach
suggests a solution to this problem by using first the 9^-operator to provide an
initial specification of the probability model of the map in terms of the <j-base,
and then the ^-operator to extend or refine the starting model leading to a
form that better represents the 5-base.