PHYSTAT 2011 Bayes and Discovery: Objective Bayesian Hypothesis Testing

Let z be the available data which are assumed to have been generated as one random observation from model Mz = {p(z |ω),z ∈ Z,ω ∈ Ω}. Often, but not always, data will consist of a random sample z = {x1, . . . ,xn} from some distribution q(x |ω), with x ∈ X ; in this case p(z |ω) = Qn i=1 q(xi |ω) and Z = X . Let θ(ω) be the vector of interest. Without loss of generality, the model may explicitly be expressed in terms of θ so thatMz = {p(z |θ,λ),z ∈ Z,θ ∈ Θ,λ ∈ Λ}, where λ is some appropriately chosen nuisance parameter vector. Let π(θ,λ) = π(λ |θ)π(θ) be the assumed prior, and let π(θ |z) be the corresponding marginal posterior distribution of θ. Appreciation of the inferential contents of π(θ |z) may be enhanced by providing both point and region estimates of the vector of interest θ, and by declaring whether or not some context suggested specific value θ0 is compatible with the observed data z (precise hypothesis testing). A large number of Bayesian estimation and hypothesis testing procedures have been proposed in the literature. We argue that their choice is better made in decision theoretical terms.