Discussion of Nested Sampling for Bayesian Computations by John Skilling

The ingredients to a Bayesian analysis comprise the sampling model {Pθ : θ ∈ Ω} for the response X, the prior Π for the parameter θ, and the observed value X = x. This is equivalent to specifying the joint model Pθ ×Π for (X, θ) and the observed value X = x. In many situations we may also have a loss function but we ignore this here as it is not material to our discussion. We refer to Pθ ×Π as the Bayesian model. The Bayesian model and the data comprise the full information available and we ask how this information is to be used in carrying out a statistical analysis. The joint model can be factored as Pθ × Π = M × Π (· |X) where M is the marginal prior predictive for X and Π (· |X) is the posterior for θ. The principle of conditional probability then says that probability statements about θ, that are initially based on the prior Π, should instead be based on the observed posterior Π (· |X = x) . What then, is the role of M in a statistical analysis? If our goal is inference about θ, then it might seem that we can ignore M and proceed directly to work with Π (· |X = x) . This may seem even preferable if it is