The geometry of exponential families
Parametric statistics concerns parametrised families of probability distributions p(θ), where the parameter θ = (θ ^{1},…, θ ^{ d }) varies over some open set in R ^{ d }. The most common example is the normal family, which is usually expressed as a family of densities p ( μ , σ ) = N ( μ , σ 2 ) = 1 2 π σ exp ( − ( x − μ ) 2 2 σ 2 ) The parameter θ in this case is the pair (μ, σ) which varies over the open subset of R ^{2} determined by μ > 0. The sample space is R and the densities are with respect to Lebesgue measure dx on R, so that as a set of probability measures the normal family is N = { p ( μ , σ ) d x | μ ∈ R , σ > 0 }