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arxiv: 2606.22687 · v1 · pith:5HFVHJSQnew · submitted 2026-06-21 · 📊 stat.ME

What is your Prior Worth? Effective Sample Size and Sample Size Planning for Gaussian Graphical Models

Giuseppe Arena , Lourens Waldorp , Maarten Marsman This is my paper

Pith reviewed 2026-06-26 09:37 UTC · model grok-4.3

classification 📊 stat.ME
keywords effective sample sizeGaussian graphical modelsWishart priorG-Wishart priorsample size planningBayesian analysisprecision matrixnetwork models
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The pith

Priors for Gaussian graphical models can now be expressed as equivalent numbers of observations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops formal pre-data measures of effective sample size for priors on Gaussian graphical models under the Wishart and G-Wishart distributions. It adapts five existing ESS estimators and aggregates them in two ways, one global via determinant ratio and one parameterwise via Cholesky factors, to put prior strength into observation units. This supports two planning tools: one that finds the sample size where data overtake the prior and one that targets conclusive Bayes factor evidence on edges. A sympathetic reader would care because network models previously lacked any principled basis for deciding how large a study must be when an informative prior is used.

Core claim

The authors formalize a pre-data effective sample size for GGMs under the Wishart and G-Wishart priors. They adapt five ESS estimators and compute each through a global determinant-ratio scheme and a parameterwise Cholesky scheme. These measures then underpin a Data-to-Prior Information Ratio that identifies when data dominate the prior and a GGM version of Bayes Factor Design Analysis that identifies the sample size needed for conclusive edge evidence. Simulations show the two procedures address complementary design goals and that the estimators respond differently to network structure.

What carries the argument

Pre-data effective sample size (ESS) for GGMs, obtained by adapting five estimators and aggregating them globally by determinant ratio or parameterwise by Cholesky decomposition.

If this is right

  • The Data-to-Prior Information Ratio identifies the smallest sample size at which the data dominate the prior.
  • The extended Bayes Factor Design Analysis identifies the sample size required for conclusive edge-based Bayes factor evidence.
  • Different ESS estimators produce systematically different planning recommendations because they vary in sensitivity to network structure and geometry.
  • The same machinery applies equally to the Wishart and the G-Wishart prior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The measures could be checked against actual posterior concentration in small-sample real-data analyses to see whether the predicted dominance occurs at the planned sizes.
  • Applying the same aggregation logic to other matrix-variate priors might reveal whether the dependence structure of GGMs is unusually difficult to express in observation units.
  • Comparing the resulting ESS values across different graphical-model families could show whether the planning tools transfer without modification.

Load-bearing premise

The five ESS estimators remain valid after adaptation to the dependence among precision-matrix entries induced by the Wishart and G-Wishart priors.

What would settle it

A simulation in which the posterior precision matrix after a sample size equal to the computed ESS does not exhibit the degree of concentration predicted by that ESS value.

Figures

Figures reproduced from arXiv: 2606.22687 by Giuseppe Arena, Lourens Waldorp, Maarten Marsman.

Figure 1
Figure 1. Figure 1: DPIR sample size planning for the toy example with [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: BFDA results for the planning edge (i, j) with ρmin = 0.210, p = 10, and ν = 100. Panel (a): Power curves β0 (red) and β1 (blue) for the planning edge as a function of n. Dotted lines mark the two planned sample sizes and the target power of 0.8. Panel (b): FNR (red) and FPR (blue) for the planning edge as a function of n. Dotted lines mark the two planned sample sizes and the target error rate of 0.05. Mu… view at source ↗
Figure 3
Figure 3. Figure 3: log(BF01) distribution across all present edges at n ⋆ H1 = 294 (top row) and n ⋆ H0 = 317 (bottom row), under H0 (left column) and H1 (right column). Lines show the median log(BF01) and shaded ribbons the interquartile range across prior predictive draws. Dark blue: edges in E +; light gray: edges below ρmin. Dashed vertical line at ρmin = 0.210; dotted horizontal lines at ± log(γ) with γ = 10 and at 0. T… view at source ↗
Figure 4
Figure 4. Figure 4: Jensen gap J = 1−ESSPR/ESSVR as a function of ν, for the Wishart (solid) and G-Wishart (dashed) priors, colored by p. The gap decreases with ν and increases with p for both priors. Jensen gap [PITH_FULL_IMAGE:figures/full_fig_p029_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Jensen gap J = 1 − ESSPR/ESSVR plotted against network density across network sizes p (columns) and prior study sizes ν (color); the dotted line marks the Jensen lower bound at 0. decreasing at the highest densities. Since J combines the two effective sample sizes, ESSVR which approaches ν from above and ESSPR which approaches ν from below, this non-monotonicity originates in how the two measures respond t… view at source ↗
Figure 6
Figure 6. Figure 6: ESS/ν ratio plotted against network density across network sizes p (columns) and prior study sizes ν (color), for the ESSVR (top) and ESSPR (bottom) estimators. The dotted horizontal line marks ESS/ν = 1, where the prior contributes as many effective observations as the prior study size. rises above 1 for sparse networks, peaks around density 0.5 − 0.8, and then decreases for denser networks. The crossover… view at source ↗
Figure 7
Figure 7. Figure 7: Network metrics normalized to [0, 1] within each (p, ν) condition, plotted against network density across network sizes p (columns) and prior study sizes ν (color). The dotted bisector marks perfect linear tracking of density. numerator matrix of each ESS measure: EΘ[I −1 (x1; Θ)] for ESSVR and V −1 (Θ) for ESSPR, and is also geometry-dependent. The log condition numbers of the denomi￾nators and the log co… view at source ↗
Figure 8
Figure 8. Figure 8: log10(n ⋆/ESSVR) plotted against prior study size ν across network sizes p (columns) and sample size plannin methods (rows), for dense (solid black) and sparse (dashed gray) priors. The dotted horizontal line marks log10 = 0, where n ⋆ = ESSVR [PITH_FULL_IMAGE:figures/full_fig_p034_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Pr(n ⋆ mis > n⋆ ref) for the BFDA planning strategy as a function of the planning edge shift ∆ρij = ρmis−ρref, across prior study sizes ν (color), network sizes p (columns). The dashed line marks Pr = 0.5 and the dotted line marks ∆ρ = 0. Bins of width 0.05; outliers removed per condition using the 1.5 × IQR rule. sample size exceeds ν, reflecting edges whose presence or absence in the prior graph is ambig… view at source ↗
Figure 10
Figure 10. Figure 10: Pr(n ⋆ mis > n⋆ ref) for the DPIR planning strategy, where n ⋆ = max(n ⋆ global, max(i,j)∈/diag n ⋆ (θij )), as a function of the planning edge shift ∆ρij = ρmis −ρref, across prior study sizes ν (color) and network sizes p (columns). The dashed line marks Pr = 0.5 and the dotted line marks ∆ρ = 0. Bins of width 0.05; outliers removed per condition using the 1.5 × IQR rule. planned sample size relative to… view at source ↗
read the original abstract

In Bayesian analysis, the prior effective sample size (ESS) expresses the information carried by a prior distribution in units of observations, quantifying how much independent information the prospective data must provide to outweigh an informative prior elicited from a previous study. For network models such as Gaussian graphical models (GGMs), the prior ESS is not straightforward to compute. The Wishart and G-Wishart priors induce dependence among the entries of the precision matrix, and their informativeness has never been expressed in an interpretable, observation-equivalent unit. As a result, researchers eliciting an informative prior for a GGM have had no principled basis for sample size planning. In this paper, we close this gap by formalizing a pre-data ESS for GGMs under the Wishart and G-Wishart priors. We adapt five ESS estimators to the GGM setting and compute each through two aggregation schemes: a global ESS measure based on a determinant ratio, and a parameterwise version based on a Cholesky decomposition. Building on these measures, we introduce two complementary planning strategies: the Data-to-Prior Information Ratio (DPIR), which determines the sample size at which the data dominate the prior, and a GGM extension of Bayes Factor Design Analysis (BFDA), which determines the sample size required for conclusive edge-based evidence. Simulation studies show that the two procedures target complementary design goals and that the ESS estimators differ systematically in their sensitivity to network structure and geometry. We conclude by outlining extensions to other graphical models, including time-dependent variants, as well as to matrix-variate mixture priors.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper formalizes a pre-data effective sample size (ESS) for Gaussian graphical models (GGMs) under Wishart and G-Wishart priors. It adapts five standard ESS estimators from the non-graphical literature, computes each via a global aggregation (determinant ratio) and a parameterwise aggregation (Cholesky decomposition), and uses the resulting measures to define the Data-to-Prior Information Ratio (DPIR) for determining when data dominate the prior and a GGM extension of Bayes Factor Design Analysis (BFDA) for edge-based evidence. Simulation studies are reported to show that the procedures target complementary goals and that the ESS estimators differ in sensitivity to network structure.

Significance. If the adapted ESS quantities correctly map prior degrees of freedom to observation-equivalent units while respecting the conditional-independence constraints of the graph, the work would close a genuine methodological gap and supply concrete, interpretable tools for prior elicitation and sample-size planning in network models. The dual aggregation schemes and the complementary DPIR/BFDA planning rules are potentially useful distinctions.

major comments (3)
  1. [Section describing adaptation of the five ESS estimators] The manuscript provides no verification that any of the five adapted ESS estimators recover the known ESS values for the complete-graph case (ordinary Wishart prior). Without this check, it is unclear whether the adaptation preserves the observation-equivalent interpretation once the precision matrix is restricted to a graph.
  2. [Section on parameterwise aggregation via Cholesky decomposition] The parameterwise (Cholesky) aggregation scheme is not shown to be invariant to vertex relabeling. Because the Cholesky factor depends on ordering and the graph induces conditional dependencies, different labelings could produce different ESS values, undermining the claim that the measure quantifies prior informativeness in stable observation units.
  3. [Section introducing DPIR and BFDA] No analytic or numerical argument is given that the global (determinant-ratio) aggregation remains consistent with the parameterwise scheme when the graph is sparse; if the two schemes diverge systematically under conditional independence constraints, the subsequent DPIR and BFDA rules rest on an ambiguous definition of prior strength.
minor comments (2)
  1. The abstract states that simulations illustrate sensitivity to network structure and geometry, but the main text should include a table or figure that directly compares the five estimators across the same set of graphs and sample sizes.
  2. Notation for the two aggregation schemes should be introduced with explicit formulas (e.g., the determinant ratio and the Cholesky-based sum) before they are used in the DPIR and BFDA definitions.

Simulated Author's Rebuttal

3 responses · 0 unresolved

Thank you for the detailed and constructive referee report. We address each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [Section describing adaptation of the five ESS estimators] The manuscript provides no verification that any of the five adapted ESS estimators recover the known ESS values for the complete-graph case (ordinary Wishart prior). Without this check, it is unclear whether the adaptation preserves the observation-equivalent interpretation once the precision matrix is restricted to a graph.

    Authors: We agree that explicit verification for the complete-graph (unrestricted Wishart) case is necessary to confirm that the adaptation preserves the standard interpretation. In the revised manuscript we will add a dedicated verification subsection (or appendix) demonstrating recovery of the known ESS values under the ordinary Wishart prior when the graph is complete. revision: yes

  2. Referee: [Section on parameterwise aggregation via Cholesky decomposition] The parameterwise (Cholesky) aggregation scheme is not shown to be invariant to vertex relabeling. Because the Cholesky factor depends on ordering and the graph induces conditional dependencies, different labelings could produce different ESS values, undermining the claim that the measure quantifies prior informativeness in stable observation units.

    Authors: This is a substantive point. The Cholesky factor is ordering-dependent. In revision we will (i) explicitly state that the parameterwise ESS is defined with respect to a fixed, user-chosen vertex ordering consistent with the supplied graph, (ii) add numerical checks across random relabelings in the simulation studies, and (iii) discuss practical recommendations for choosing or stabilizing the ordering. revision: partial

  3. Referee: [Section introducing DPIR and BFDA] No analytic or numerical argument is given that the global (determinant-ratio) aggregation remains consistent with the parameterwise scheme when the graph is sparse; if the two schemes diverge systematically under conditional independence constraints, the subsequent DPIR and BFDA rules rest on an ambiguous definition of prior strength.

    Authors: We will strengthen this section by adding targeted numerical comparisons of the two aggregation schemes on sparse graphs within the existing simulation framework. These comparisons will quantify agreement/divergence under conditional independence and will be used to clarify the conditions under which DPIR and BFDA remain well-defined. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper adapts five existing ESS estimators from the non-graphical literature to the Wishart and G-Wishart priors on GGMs, then applies determinant-ratio and Cholesky aggregation schemes to produce observation-equivalent units. No equations or definitions in the provided abstract or description reduce the new ESS quantities to fitted parameters or to the target result by construction. The central formalization is presented as an extension of independent prior work, with simulation studies offered as external checks. This meets the criteria for a self-contained derivation against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no specific free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.1-grok · 5818 in / 989 out tokens · 18916 ms · 2026-06-26T09:37:36.781516+00:00 · methodology

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