Exponential families are a particular class of statistical manifolds which are important in statistical inference, and which appear very frequently in statistics. For example, the set of normal distributions, with mean $\mu$ and deviation $\sigma$ forms a 2-dimensional exponential family.

In this lecture, we show that the tangent bundle of an exponential family is naturally a Kähler manifold. This observation, although simple, is crucial in that it leads to the formalism of quantum mechanics in its geometrical form, i.e. based on the Kähler structure of the complex projective space.

Many questions related to this "statistical Kähler geometry" are discussed, and a close connection with representation theory is observed.

Examples of physical relevance are treated in detail. For example, it is shown that the spin of a particle can be entirely understood by means of the usual binomial distribution.

This lecture centers on the mathematical foundations of quantum mechanics, and on the question of its potential generalization through its geometrical formulation.