Loss Models: From Data to Decisions, 4th Edition

15.3 Conjugate prior distributions and the linear exponential family

The linear exponential family introduced in Section 5.4 is also useful in connection with Bayesian analysis, as is demonstrated in this section.

In Example 15.8 it turned out the posterior distribution was of the same type as the prior distribution (gamma). A definition of this concept follows.

Definition 15.17 A prior distribution is said to be a conjugate prior distribution for a given model if the resulting posterior distribution is from the same family as the prior (but perhaps with different parameters).

The following theorem shows that, if the model is a member of the linear exponential family, a conjugate prior distribution is easy to find.

Theorem 15.18 Suppose that given = θ the random variables X₁, …, X_n are i.i.d. with pf

where has pdf

where k and μ are parameters of the distribution and c(μ, k) is the normalizing constant. Then the posterior pf π_θ|x(θ|X) is of the same form as π(θ).

Proof: The posterior distribution is

which is of the same form as π(θ) with parameters

EXAMPLE 15.10

Show that, for the ...