chapter  3
34 Pages

Statistical Manifolds

WithMichael K. Murray, John W. Rice

Any parametrised family of probability distributions, P = { p ( x ,   θ ) } with parameter θ running over some open subset of R r , is automatically a manifold, in which the probability distributions are the points of the manifold and the parameters are co-ordinate functions. However, this is not to say that the family can be regarded as some definite surface, or hypersurface, on which the parameters play the role of co-ordinates. A manifold can be realised as a surface or hypersurface in an infinite variety of ways. Consider, for example the α-embeddings of Amari (1985): F α ( p ) = { 2 1 − α p 1 − α 2 , α ≠ 1 log ( p ) , α = 1 which map P into R Ω, the space of random variables.