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On a conjecture of Roverato regarding G-Wishart normalising constants

Ching Wong, Giusi Moffa, Jack Kuipers

TL;DR

This paper investigates Roverato's conjecture that the G-Wishart normalising constant $\mathscr C_\mathcal{G}(\delta,D)$ can be rewritten using identity-scale constants. It recasts the conjecture in an equivalent form via the PD-completion $D^{\mathcal G}$ and the Isserlis matrix $\mathrm{Iss}_{\mathcal G}(D)$, and analyzes a one-edge deletion setup to derive a gamma-function based ratio that the conjecture would force. A concrete counterexample (a 4-cycle vs a 3-edge path) demonstrates that the conjectured equality fails in general, though a related ratio approximation remains a good leading-order estimate, especially for large $\delta$ or data size. The work clarifies the limits of Roverato's conjecture and highlights the practical utility of the induced ratio approximation in Bayesian computation, while leaving open questions about higher-order corrections and data-invariant closed forms.

Abstract

The evaluation of G-Wishart normalising constants is a core component for Bayesian analyses for Gaussian graphical models, but remains a computationally intensive task in general. Based on empirical evidence, Roverato [Scandinavian Journal of Statistics, 29:391--411 (2002)] observed and conjectured that such constants can be simplified and rewritten in terms of constants with an identity scale matrix. In this note, we disprove this conjecture for general graphs by showing that the conjecture instead implies an independently-derived approximation for certain ratios of normalising constants.

On a conjecture of Roverato regarding G-Wishart normalising constants

TL;DR

This paper investigates Roverato's conjecture that the G-Wishart normalising constant can be rewritten using identity-scale constants. It recasts the conjecture in an equivalent form via the PD-completion and the Isserlis matrix , and analyzes a one-edge deletion setup to derive a gamma-function based ratio that the conjecture would force. A concrete counterexample (a 4-cycle vs a 3-edge path) demonstrates that the conjectured equality fails in general, though a related ratio approximation remains a good leading-order estimate, especially for large or data size. The work clarifies the limits of Roverato's conjecture and highlights the practical utility of the induced ratio approximation in Bayesian computation, while leaving open questions about higher-order corrections and data-invariant closed forms.

Abstract

The evaluation of G-Wishart normalising constants is a core component for Bayesian analyses for Gaussian graphical models, but remains a computationally intensive task in general. Based on empirical evidence, Roverato [Scandinavian Journal of Statistics, 29:391--411 (2002)] observed and conjectured that such constants can be simplified and rewritten in terms of constants with an identity scale matrix. In this note, we disprove this conjecture for general graphs by showing that the conjecture instead implies an independently-derived approximation for certain ratios of normalising constants.

Paper Structure

This paper contains 5 sections, 20 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. An equivalent form of the conjecture
  3. The conjecture implies the ratio estimate
  4. A counter example
  5. Conclusions

Figures (2)

  • Figure 1: For the first ten positive integers $\delta$, we compare the correct ratio from Equation (\ref{['eq:true']}) and the approximation of Equation (\ref{['eq:approx']}).
  • Figure 2: (a) The graphs $\mathcal{G}_1$, $\mathcal{G}_2$, $\mathcal{G}_3$ (from left to right), are the non-chordal graphs for Fisher's Iris Virginica dataset. (b) Violin plots of the estimates of the values of $\log(\mathscr C_{\mathcal{G}_j}(53, U+I_4))$, where $j = 1,2,3$, using Monte Carlo integration Atay-Kayismw19 with $1000$ samples and for 200 different seeds. The horizontal lines represent the values obtained using Roverato's conjecture and the exact one-dimensional integral in GWishart.

Theorems & Definitions (1)

  • Conjecture 1: Roverato