Objective Bayesian Inference for a Generalized Marginal Random Effects Model

Abstract

An objective Bayesian inference is proposed for the generalized marginal random effects model $p\left(\mathbf{x}\right|\mu ,{\sigma }_{\lambda })=f({(\mathbf{x}-\mu \mathbf{1})}^T{(\mathbf{V}+{\sigma }_{\lambda }^2\mathbf{I})}^{-1}(\mathbf{x}-\mu \mathbf{1}))/\sqrt{\det (\mathbf{V}+{\sigma }_{\lambda }^2\mathbf{I})}$. The matrix $\mathbf{V}$ is assumed to be known, and the goal is to infer $\mu$ given the observations $\mathbf{x}={(x_1,\dots ,x_n)}^T$, while ${\sigma }_{\lambda }$ is a nuisance parameter. In metrology this model has been applied for the adjustment of inconsistent data $x_1,\dots ,x_n$, where the matrix $\mathbf{V}$ contains the uncertainties quoted for $x_1,\dots ,x_n$.

We show that the reference prior for grouping $\{\mu ,{\sigma }_{\lambda }\}$ is given by $\pi (\mu ,{\sigma }_{\lambda })\propto \sqrt{{\mathbf{F}}_{22}}$, where ${\mathbf{F}}_{22}$ denotes the lower right element of the Fisher information matrix $\mathbf{F}$. We give an explicit expression for the reference prior, and we also prove propriety of the resulting posterior as well as the existence of mean and variance of the marginal posterior for $\mu$. Under the additional assumption of normality, we relate the resulting reference analysis to that known for the conventional balanced random effects model in the asymptotic case when the number of repeated within-class observations for that model tends to infinity.

We investigate the frequentist properties of the proposed inference for the generalized marginal random effects model through simulations, and we also study its robustness when the underlying distributional assumptions are violated. Finally, we apply the model to the adjustment of current measurements of the Planck constant.