Consider a distribution for which the p.d.f. or the p.f. is f(x|θ), where θ belongs to some parameter space It is said that the family of distributions obtained by letting θ vary over all values in is an exponential family, or a Koopman-Darmois family, if f (x|θ) can be written as follows for θ ∈ and all values of x:
f (x|θ) = a(θ)b(x) exp[c(θ) d(x)].
Here a(θ) and c(θ) are arbitrary functions of θ, and b(x) and d(x) are arbitrary functions of x. Let
For each (α, β) ∈ H, let
and let ψ be the set of all probability distributions that have p.d.f.’s of the form ξα,β(θ) for some (α, β) ∈ H.
a. Show that ψ is a conjugate family of prior distributions for samples from f (x|θ).
b. Suppose that we observe a random sample of size n from the distribution with p.d.f. f (x|θ). If the prior p.d.f. of θ is ξα0,β0, show that the posterior hyperparameters are
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