chapter  1
24 Pages

The geometry of exponential families

WithMichael K. Murray, John W. Rice

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 }