The paper first reviews the existing mathematical theory for the estimation of a random field T, with known covariance C, from a finite vector of observations, related to T by linear functionals, in the framework of a Bayesian approach. In particular Section 2 and Section 3 the equivalence between ordinary collocation formulas, their explanations in terms of generalized random fields and the full probabilistic picture is demonstrated. Then the more general problem of estimating from data both T and the covariance C is tackled; in Section 4 it is shown how to reduce the problem by using prior invariance principles and a prior vague information on the regularity of the field T. In this case it is shown how to construct the posterior distribution of the unknowns. Then in Section 5 the theory of logarithmic derivatives of infinite dimensional distributions is recalled and in Section 6 the corresponding Maximum a-Posteriori equations are constructed. A theorem of existence of at least one solution is proved too. Conclusions follow.

The estimation theory for random fields in the bayesian context: a contribution from geodesy

SANSO', FERNANDO;VENUTI, GIOVANNA
2004-01-01

Abstract

The paper first reviews the existing mathematical theory for the estimation of a random field T, with known covariance C, from a finite vector of observations, related to T by linear functionals, in the framework of a Bayesian approach. In particular Section 2 and Section 3 the equivalence between ordinary collocation formulas, their explanations in terms of generalized random fields and the full probabilistic picture is demonstrated. Then the more general problem of estimating from data both T and the covariance C is tackled; in Section 4 it is shown how to reduce the problem by using prior invariance principles and a prior vague information on the regularity of the field T. In this case it is shown how to construct the posterior distribution of the unknowns. Then in Section 5 the theory of logarithmic derivatives of infinite dimensional distributions is recalled and in Section 6 the corresponding Maximum a-Posteriori equations are constructed. A theorem of existence of at least one solution is proved too. Conclusions follow.
2004
V Hotine-Marussi Symposium on Mathematical Geodesy
354021979X
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11311/563118
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