Statistical Geometry
Dorje C. Brody, Lane P. Hughston · arXiv (Cornell University) · 1997
A statistical model M is specified by a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square-integrable functions. More precisely, by consideration of the square-root density function we can regard M as a submanifold of the unit sphere S in a real Hilbert space H. Therefore, H effectively embodies the `state space' of the probability distributions, and the geometry of the given statistical model can be described in terms of the embedding of M in S. The geometry in question is characterised by a natural Riemannian metric (the Fisher-Rao metric), and as a consequence various aspects of classical statistical inference can be formulated in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramér-Rao and Bhattacharyya bounds, described in terms of the geometry of the underlying real Hilbert space. The statistical model M can then be specialised to the case of a submanifold of the state space of a quantum mechanical system. This can be pursued by introducing a compatible complex structure on the underlying real Hilbert space, thus allowing the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds to quantum statistical estimation is shown to lead to higher order corrections to the Heisenberg uncertainty relations.