Design and Structure Dependent Priors for Scale Parameters in Latent Gaussian Models

Bayesian inference in latent Gaussian models necessitates the specification of prior distributions for scale parameters, which govern the behavior of model components. This task is particularly delicate and many contributions in the literature are devoted to the topic. We show that the scale parameter plays a crucial role in determining the prior variability of the model components, which is influenced by factors such as correlation structure, design matrices, and potential linear constraints. This intricate relationship adds complexity, making it difficult to interpret and compare priors across diverse applications. To tackle this challenge, we propose a novel approach for prior specification based on the theory of distribution of quadratic forms. Our strategy involves the use of design and structure-dependent (DSD) priors, which ensure a consistent interpretation across diverse applications. By introducing a single parameter that governs the prior variability of the linear predictor, we simplify the process of prior specification, making it more manageable and interpretable. We derive analytical expressions for DSD priors on scale parameters and establish conditions that guarantee their existence. To demonstrate the efficacy of our proposed prior elicitation strategy, we conduct a simulation study, examining the sampling properties of the estimators. Additionally, we explore several real data applications to investigate prior sensitivity and the allocation of explained variance among model components.