Introduction¶
The single-parameter exponential family is a class of distributions that can be expressed as:
f(y; θ, 𝜙) = exp{(yθ - b(θ))/a(𝜙) + c(y,𝜙)}.
for which 𝜙
is assumed to be known.
The definition of the functions a(.)
, b(.)
, and c(.)
determines a
probabilistic distribution having the canonical parameter θ
. The expectation of
y
, denoted here by 𝜇
, determines the value of θ
via the following relation:
b'(θ) = 𝜇
Still, the value 𝜇
is often set indirectly via the natural parameter η
, which
relates to each other through a link function g(.)
:
η = g(𝜇)
If g(.)
is the so-called canonical function, we have the desirable equality:
θ = η