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130 Modern Spatiotemporal Geostatistics — Chapter 6

Posterior Operators for Other Forms
of Soft Data

Working along the lines of the proofs of Propositions 6.1 and 6.2, the validity
of the following two propositions can be demonstrated.

PROPOSITION 6.3: Assume that the specificatory knowledge consists
of the hard data (Eq. 3.30, p. 84) and the probabilistic (soft) data of
Equation 3.34 (p. 86). Then, the posterior operator is given by

where R 1 is the real line, and /(() is the domain of x aoft determined by

PROPOSITION 6.4: Assume that the specificatory knowledge S con-
sists of the hard data (Eq. 3.30) and the probabilistic (soft) data of
Equation 3.35 (p. 86). Then,

where with
Next, Propositions 6.3 and 6.4 above are illustrated with the help of a few
examples.
EXAMPLE 6.3: Let x soft = (Xi, Xa) such thai

This is shown diagrammatically in Figure 6.2.
In view of Equation 6.10, the posterior operator is given by

As usual, the posterior pdf is as follows: