The estimation theory for random fields in the Bayesian context; a contribution from Geodesy
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 in Section 2 and Section 3 the equivalence between ordinary collocation formulas, their explaination 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 its 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.
KeywordsHilbert Space Posterior Distribution Random Field Covariance Function Bayesian Approach
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