# 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