Scale Parameters: A Bayesian Treatment

Abstract

Summary A definition of a scale parameter ϕ is given which includes the usual definition of a scale parameter in the exponential family. Some suggestions for the estimation of ϕ and approximations to the marginal posterior distribution of θ, a vector-valued parameter of primary interest, are made. In particular, the possibility of a χ2 approximation to the posterior distribution of ϕ, leading to h.p.d. regions for θ based on the F distribution, is explored. Two examples are discussed, where observations are from a normal and gamma distribution.