Hierarchical Bayesian Whittaker Graduation

The Whittaker method of graduation has been known and used for a long time and has remained popular due to its possession of a number of ideal properties. They include being nonparametric and having an easy to understand foundation. The latter means that it makes sense and thus the user of the method has a good idea of what it can and cannot do. As well, there is a statistical derivation available that uses Bayesian notions. A problem with the derivation is that it is more intuitive than precise and as such does not provide a useful frame of reference for the graduator. Regardless of the point of view, the graduation cannot be completed until the smoothing parameter is selected and this has always relied on the judgment of the analyst. In this paper, three tasks will be undertaken. The first is to replace the ad-hoc Bayesian derivation of the method with a formal Bayesian specification. The second is to show that with this specification it is possible to complete the graduation without making an arbitrary selection of the smoothing parameter. The third is to provide a Monte Carlo Bayesian approach for the incorporation of constraints in the graduated values. The ideas will be illustrated with a numerical example.