Page 146 - Modern Spatiotemporal Geostatistics

Analytical Expressions of the Posterior Operator 127

/-domain corresponds to the subset of data points p i at which interval (soft)
data rather than actual observations are available.

Below we examine a few more cases of ^-operators. Unlike Proposition
6.1 in which the probability space of the prior stage was the same as that of
the meta-prior stage (conditioning knowledge was known with certainty), in the
following proposition the probability space of the soft data at the meta-prior
stage is different than the probability space of the prior stage (the conditioning
knowledge is uncertain, and the existence of a new probability space will be
denoted by the subscript S)-

PROPOSITION 6.2: Assume that the specificatory knowledge S con-
sists of the hard data (Eq. 3.30, p. 84) and the probabilistic (soft) data
(Eq. 3.33, p. 86). Then, the posterior operator is given by

where Z is the partition function.
Proof: Let I = (I mh+i,..., I m) be the domain of the soft data vector x soft
!
such that Xi e /« (i = vn-h + , • • • , in)- The notation x so^ e / denotes an
event with respect to the prior probability law P§[x soft e I], i.e., before S
is taken into consideration. The notation x soft e I(S), on the other hand,
denotes an event with respect to the new probability law Pg \x soft & I(S)],
i.e., after acquiring S. Let us define

where see Equation 3.33, so that

e 7andx so/j € 1(5) within the probability
Herein, for simplicity, the\ soft
laws will simply be replaced by I and 1(5). Each interval Jj of / is partitioned
into a number of mutually exclusive and exhaustive sub-intervals
Ui = 1,..., Ni. By choosing one jj jUi from each 7j, we can define all possible