Theory of Statistics - George Mason University

342 4 Bayesian Inference
r(FΘ, T) = L(θ, T(x))dFX|θ(x)dFΘ(θ). (4.39)
Θ X
exists, we call a rule that minimizes it a generalized Bayes action.
In Example 4.9 on page 355, we determine the Bayes estimator of the mean
in a normal distribution when the prior is uniform over the real line, which
obviously is an improper prior.
Limits of Bayes Actions
Another variation on Bayes actions is the limit of the Bayes action as some
hyperparameters approach fixed limits. In this case, we have a sequence of
prior PDFs, {F (k)
Θ : k = 1, 2, . . .}, and consider the sequence of Bayes actions,
δ (k) (X) that result from these priors. The limit limk→∞ δ (k) (X) is called a
limiting Bayes action. The limiting prior, limk→∞ F (k)
Θ , may not be a PDF.
In Example 4.6 on page 352, we determine a limiting Bayes action, given
in equation (4.47).
4.2.5 Choosing Prior Distributions
It is important to choose a reasonable prior distribution that reflects prior
information or beliefs about the phenomenon of interest.
Families of Prior Distributions
Various families of prior distributions can provide both flexibility in representing
prior beliefs and computational simplicity. Conjugate priors, as in
Examples 4.2 through 4.5, are often very easy to work with and to interpret.
In many cases, a family of conjugate priors would seem to range over
most reasonable possibilities, as shown in Figure 4.1 for priors of the binomial
parameter.
Within a given family, it may be useful to consider ones that are optimal
in some way. For example, in testing composite hypotheses we may seek a
“worst case” for rejecting or accepting the hypotheses. This leads to consideration
of a “least favorable prior distribution”. We may also wish to use a
prior that reflects an almost complete lack of prior information.. This leads
to consideration of “noninformative priors”, or priors with maximum entropy
within a given class.
If the priors are restricted to a particular class of distributions, say Γ,
we may seek an action whose worst risk with respect to any prior in Γ is
minimized, that is, we may see an action δ that yields
inf
δ sup r(P, δ).fΘ(θ) ∝ |I(θ)|
P ∈Γ
1/2 . (4.40)
Such an action is said to be Γ-minimax, usually written as gamma-minimax.
Clearly, any minimax Bayes action is gamma-minimax Bayes with respect to
the same loss function.
Theory of Statistics c○2000–2013 James E. Gentle