Differential geometry has found fruitful application in statistical inference. In particular, Amari's (1990) expected geometry is used in higherorder asymptotic analysis and in the study of sufficiency and ancillarity. However, we can see three drawbacks to the use of a differential geometric approach in econometrics and statistics more generally. First, the mathematics is unfamiliar and the terms involved can be difficult for the econometrician to appreciate fully. Secondly, their statistical meaning can be less than completely clear. Finally, the fact that, at its core, geometry is a visual subject can be obscured by the mathematical formalism required for a rigorous analysis, thereby hindering intuition. All three drawbacks apply particularly to the differential geometric concept of a non-metric affine connection.

The primary objective of this chapter is to attempt to mitigate these drawbacks in the case of Amari's expected geometric structure on a full exponential family. We aim to do this by providing an elementary account of this structure that is clearly based statistically, accessible geometrically and visually presented.

Statistically, we use three natural tools: the score function and its first two moments with respect to the true distribution. Geometrically, we are largely able to restrict attention to tensors; in particular, we are able to avoid the need formally to define an affine connection. To emphasise the visual foundation of geometric analysis we parallel the mathematical development with graphical illustrations using important examples of full exponential families.