Statistical Manifolds
Michael K. Murray, John W. Rice · 2017
Any parametrised family of probability distributions, P = { p ( x , θ ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315141268/7ec67f00-5aa3-4444-a5a8-38b4589858e0/content/eq330.tif"/> 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 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315141268/7ec67f00-5aa3-4444-a5a8-38b4589858e0/content/eq331.tif"/> which map P into R Ω , the space of random variables.