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

Aldo Gardini, Fedele Greco, Carlo Trivisano

TL;DR

The paper tackles the challenge of eliciting priors for scale parameters in Latent Gaussian Models when prior information must be translated through complex design and correlation structures. It introduces Design and Structure Dependent (DSD) priors, deriving a closed-form prior on $\sigma^2$ that yields a target marginal distribution for the random-effects sampling variance $V_{\nu}$ independent of $\mathbf{K}_{\nu}$ and $\mathbf{Z}$ by leveraging Quadratic Forms and Mellin-transform techniques; special cases recover standard B2 and Half-t priors. The authors extend the framework to intrinsic priors for IGMRF components, provide practical prior-elicitation guidelines (default $p=\tfrac12$, $q=\tfrac32$, and a data-driven $\pi_0$), and demonstrate via simulations and two real-data applications (Munich rental, Tokyo rainfall) that DSD priors stabilize inference when the model structure is complex and allow transparent sensitivity analysis. The approach yields coherent variance allocation across model components, improves robustness for small basis sizes, and offers a practical, architecture-aware tool for priors in LGMs with broad applicability to spatial, temporal, and semi-parametric settings.

Abstract

Many common correlation structures assumed for data can be described through latent Gaussian models. When Bayesian inference is carried out, it is required to set the prior distribution for scale parameters that rules the model components, possibly allowing to incorporate prior information. This task is particularly delicate and many contributions in the literature are devoted to investigating such aspects. We focus on the fact that the scale parameter controls the prior variability of the model component in a complex way since its dispersion is also affected by the correlation structure and the design. To overcome this issue that might confound the prior elicitation step, we propose to let the user specify the marginal prior of a measure of dispersion of the model component, integrating out the scale parameter, the structure and the design. Then, we analytically derive the implied prior for the scale parameter. Results from a simulation study, aimed at showing the behavior of the estimators sampling properties under the proposed prior elicitation strategy, are discussed. Lastly, some real data applications are explored to investigate prior sensitivity and allocation of explained variance among model components.

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

TL;DR

The paper tackles the challenge of eliciting priors for scale parameters in Latent Gaussian Models when prior information must be translated through complex design and correlation structures. It introduces Design and Structure Dependent (DSD) priors, deriving a closed-form prior on that yields a target marginal distribution for the random-effects sampling variance independent of and by leveraging Quadratic Forms and Mellin-transform techniques; special cases recover standard B2 and Half-t priors. The authors extend the framework to intrinsic priors for IGMRF components, provide practical prior-elicitation guidelines (default , , and a data-driven ), and demonstrate via simulations and two real-data applications (Munich rental, Tokyo rainfall) that DSD priors stabilize inference when the model structure is complex and allow transparent sensitivity analysis. The approach yields coherent variance allocation across model components, improves robustness for small basis sizes, and offers a practical, architecture-aware tool for priors in LGMs with broad applicability to spatial, temporal, and semi-parametric settings.

Abstract

Many common correlation structures assumed for data can be described through latent Gaussian models. When Bayesian inference is carried out, it is required to set the prior distribution for scale parameters that rules the model components, possibly allowing to incorporate prior information. This task is particularly delicate and many contributions in the literature are devoted to investigating such aspects. We focus on the fact that the scale parameter controls the prior variability of the model component in a complex way since its dispersion is also affected by the correlation structure and the design. To overcome this issue that might confound the prior elicitation step, we propose to let the user specify the marginal prior of a measure of dispersion of the model component, integrating out the scale parameter, the structure and the design. Then, we analytically derive the implied prior for the scale parameter. Results from a simulation study, aimed at showing the behavior of the estimators sampling properties under the proposed prior elicitation strategy, are discussed. Lastly, some real data applications are explored to investigate prior sensitivity and allocation of explained variance among model components.
Paper Structure (18 sections, 2 theorems, 51 equations, 11 figures, 2 tables)

This paper contains 18 sections, 2 theorems, 51 equations, 11 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Hyperpriors on Scale Parameters
  3. Prior Distribution: the Beta Distribution of the $2^{\text{nd}}$ Kind
  4. The Impact of Structure and Design
  5. Other Approaches: Splitting the Total Variability
  6. Design and Structure Dependent Priors
  7. Conditional and Marginal Priors on $V_{{\nu}_j}$
  8. Derivation of the DSD Prior
  9. The Case of Intrinsic priors
  10. Prior Elicitation
  11. Decomposition of the Linear Predictor Sampling Variance
  12. Simulation Exercise
  13. Applications
  14. Munich Rental Data
  15. Tokyo Rainfall Data
  16. ...and 3 more sections

Key Result

Proposition 3.1

Let $X$ and $Y$ be two random variables such that Then, $X\sim{_2}\mathcal{F}_0(\alpha,\beta/b,p,q)$ where $\alpha$, $p$ and $q$ are shape parameters and the scale parameter $\beta/b$ is the ratio of the scale parameters of the mixed distributions. The density of $X$ is:

Figures (11)

  • Figure 1: From top to bottom: CDFs of $V_\nu|\sigma^2=1$ (dashed line) and $\mathcal{V}|\sigma^2=1$ (solid line). CDFs and PDFs of implied priors on $\sigma$ with a B2 prior or a DSD prior for $\sigma^2$.
  • Figure 2: ARMSE and frequentist coverage with respect to the number of bases $m$ for the estimates under the considered priors and $\rho=0.25$.
  • Figure 3: Monte Carlo distributions of AEs when $m=5$.
  • Figure 4: ARMSE and frequentist coverage with respect to the number of bases $m$ for the estimates under the considered priors.
  • Figure 5: $\mathbb{E}[f(\mathbf x_j)|\mathbf{y}]$ of model components for complete and reduced datasets (q=1.5).
  • ...and 6 more figures

Theorems & Definitions (4)

  • Proposition 3.1: $_2\mathcal{F}_0$-distribution
  • proof
  • Theorem 3.1: DSD prior
  • proof