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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.
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