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Popular Methods in the Light of Modern Geostatistics 231
estimators of X(p k) at a point p k (k ^ i) is the conditional mean
The derivation of specific analytical expressions for the estimator (Eq. 12.1)
depends on the probability law of the S/TRF. Consider, e.g., the case of a
Gaussian S/TRF. Then, Equation 12.1 reduces to a linear estimator, which is
optimal among all MMSE estimators. Typically, such linear MMSE estimators
are expressed in terms of the hard data, i.e.,
where A is a vector of weights associated with the data points and involving
the space/time mean and covariance functions (see, e.g., ordinary and simple
kriging; Cressie, 1991).
From the BME perspective, if the general knowledge is limited to the
mean and covariance functions and the specificatory knowledge includes only
hard data, i.e.,
the posterior pdf is Gaussian. The Gaussian pdf is symmetric and the
BMEmode estimate is, by definition, the conditional mean, i.e., it is the same
as the MMSE estimate (the BME estimator in this case is linear, which is also
the case of the MMSE estimator). By summarizing the above discussion we
can write that
A direct consequence of the preceding analysis is the following proposition.
PROPOSITION 12.1: For a Gaussian S/TRF, the MMSE estimate ob-
tained on the basis of the physical knowledge (Eq. 12.3) coincides with
the BMEmode estimate derived using the same physical knowledge.
For non-Gaussian S/TRF, the two estimation approaches generally give
different results. Most MMSE techniques still are based on the first two
moments, although they do not offer an adequate characterization of the non-
Gaussian law. BME, on the other hand, recognizes the need to involve higher
order moments in the analysis and does it in a rigorous fashion.
COMMENT 12.1 : A n interesting consequence o f Proposition 12.1 is that
while MMSE by itself may lack a meaningful epistemic explanation (rather,
its use is justified solely on the basis of a suitably favorable track record),
it can, nevertheless, acquire such an explanation within the context of the
general BME analysis.