Maximum probability interpolant

A corollary of this result is that both the radial basis function method and the kriging method are maximum probability interpolants. 

A convenient and popular summary of such a posterior distribution is the maximum probability interpolant (known as the MAP, maximum a posteriori probability estimate). If this is calculated using the calculus of variations, then a minimisation problem, similar to that of the Tikhonov methods is obtained. In the Bayesian formulation however, the free parameters need less ad hoc arguments for their assignment and have a clearer interpretation. 

The link between kriging, radial basis functions and maximum probability interpolants could be investigated in a much deeper way than in Section 4. The huge effort to analyse and develop radial basis function methods would be made more valuable if the participants in the growing radial basis function literature were more aware of the need for statistical considerations in the scattered data problem. Deterministic approaches to problems with sparse data are not applicable in most of the problems encountered in the geosciences. 

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