MSI Weekly Bulletin - Week starting Monday, 5 April 2004

It is well known that the Gauss-Newton algorithm for solving nonlinear least squares problems is a special case of the scoring algorithm for maximizing log likelihoods. What has received less attention is that the computation of the current correction in the scoring algorithm in both line search and trust region forms can be cast as a linear least squares problem. This is an important observation both because it provides a general framework for likelihood computations which accords with current computational orthodoxy, and because it can be seen as underpinning computational procedures which have been developed for particular classes of likelihood problems (for example generalised linear models). Aspects of this orthodoxy as it affects considerations such as convergence and effectiveness will be reviewed.

In 1921 Johann Radon published a proof of Helly's convexity theorem, using this lemma (aka Radon's Theorem): Any set of d+2 points in Rd can be split in two parts in such a way that the intersections of their convex hulls is nonempty (The convex hull of A is the intersection of all the convex sets in Rd containing A). In 1964 I proved that with (p-1) (d+1) + 1 points, there will be a similar splitting in p parts. This result has led to several generalizations and still open problems involving geometry, topology and finite set theory. The aim of the talk is to present Radon's proof, sketch how a result by Bernhard Neumann (1945) led to my proof and describe some of the ensuing developments.