Page 114 - Modern Spatiotemporal Geostatistics

The Epistemic Paradigm 95

of integrating scientific and technological progress with philosophical ideals

and humanistic values.

Integration or posterior stage
At the integration or posterior stage, the new probability function is relative
to the total knowledge 3C, i.e.,

Equation 4.3 means that "the probability of a mapXfc given the total knowledge
base f^ = § U S is p'." Equation 4.3 offers a measure of the credibility or
assertibility of the proposition "if 3£, then \ k," i.e., it asserts a logical relation
between knowledge f^and the mapxt- Given 3£, the probability function (Eq.
4.3) may provide a measure of the relative quantity of random field realizations
in which \ k occurs over all possible realizations. Note that while at the prior
stage the probability (Eq. 4.1) refers to the whole domain (including data and
estimation points), i.e., p map = (p dato.i Pk)< tne probability (Eq. 4.3) of the
posterior stage includes only estimation points p k.
EXAMPLE 4.4: A well-known situation of specificatory knowledge processing
in geostatistics is conditional simulation which—by incorporating a set of
measurements—is of much greater predictive value than unconditional simula-
tion discussed in Example 4.2 above.

The probability functions (Eqs. 4.1 and 4.3) assume a connection between
mapping predictions and the available knowledge. In other words, the probabil-
ity is epistemic, supported by empirical data and related to inductive evidence.
While we seek posterior predictions that are highly probable, we nevertheless
want them to achieve this probability on the basis of total knowledge and not
on general knowledge alone. As already mentioned, in the mapping context
the specificatory knowledge S refers to the Xdat a> which for this reason is
sometimes denoted as Xdata(S)-
The analysis above has left us with a final issue to be considered within
the epistemic framework of modern spatiotemporal geostatistics; namely, how
should we process knowledge of the prior and meta-prior stages to the integra-
tion stage. There are various ways to do this. A particularly efficient way is
by means of the knowledge processing rule suggested by the following funda-
mental proposition.

PROPOSITION 4.1: The posterior mapping probability (Eq. 4.3) is
related to the prior mapping probability (Eq. 4.1) by means of the rela-
tionship

whre
Proof : It is valid that