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

This paper addresses the lack of principled sample size planning for Gaussian graphical models by formalizing a prior effective sample size under Wishart and G-Wishart priors through five adapted estimators, thereby enabling researchers to determine the data size required to either dominate prior information or achieve conclusive edge-based evidence.

Original authors: Giuseppe Arena, Lourens Waldorp, Maarten Marsman

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Giuseppe Arena, Lourens Waldorp, Maarten Marsman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery about how different variables in a system (like stress, sleep, and coffee consumption) are connected. You have two sources of information:

Your "Prior" Knowledge: This is what you already believe based on a previous study. Maybe a colleague did a small study last year and found a strong link between coffee and sleep. You want to use their findings to help you.
Your New Data: This is the fresh evidence you are about to collect.

The problem is: How much is your "Prior" knowledge actually worth?

If your previous study was tiny (say, only 5 people), but you treat it as if it were a massive study (5,000 people), your new data might get drowned out. Your conclusions would just be a repeat of the old, shaky study. But if you treat a massive, high-quality prior study as if it were just a hunch, you might ignore valuable insights.

This paper is about building a ruler to measure exactly how many "people" (observations) your prior knowledge is worth.

The Core Problem: The "Black Box" of Network Models

In the world of statistics, specifically for Gaussian Graphical Models (GGMs) (which are fancy maps showing how variables connect), researchers use a mathematical tool called a Wishart or G-Wishart distribution to represent their prior knowledge.

Think of these distributions as a "black box" containing a complex web of beliefs. While the box has a setting called "degrees of freedom" (let's call it $\nu$ ), which sounds like it tells you the sample size, it's actually misleading. Because the variables in these networks are all tangled together (dependent on each other), the "real" amount of information inside the box isn't just $\nu$ . It's a messy, tangled knot that is very hard to untangle and count.

Without a way to count this knot, researchers were flying blind. They didn't know if they needed to collect 100 new data points or 1,000 to make sure their new data was stronger than their old beliefs.

The Solution: The "Effective Sample Size" (ESS) Ruler

The authors created a new way to measure this "Prior Worth." They call it the Prior Effective Sample Size (ESS).

Instead of just looking at the setting $\nu$ , they developed five different "rulers" (estimators) to measure the information inside the black box.

The Analogy: Imagine you have a jar of marbles. Some are red, some are blue, and they are glued together in weird clusters. You want to know how many individual marbles you have.
- Some rulers just count the total weight of the jar (ignoring the glue).
- Other rulers try to untangle the clusters to count the actual marbles.
- The authors found that different rulers give slightly different answers depending on how "glued together" (dependent) the network is.

They found that for some networks, the prior is worth more than the setting $\nu$ suggests, and for others, it's worth less.

The Two-Step Strategy for Planning Your Study

Once you know how many "marbles" (observations) your prior is worth, the paper gives you two strategies to decide how big your new study needs to be.

Strategy 1: The "Data vs. Prior" Tug-of-War (DPIR)

Goal: Make sure your new data is strong enough to overpower your old beliefs.
The Metaphor: Imagine a tug-of-war. On one side is your Prior (your old study). On the other side is your New Data.

If the Prior is too heavy, the rope doesn't move, and your new study just confirms what you already "knew."
The authors' method calculates exactly how many new people you need to recruit so that the "New Data" team pulls harder than the "Prior" team.
They offer two ways to measure this:
- Global: Is the average new data stronger than the prior?
- Parameterwise: Is the new data stronger than the prior for every single connection in the network? (This is the stricter, safer option).

Strategy 2: The "Edge Detective" (BFDA)

Goal: Make sure you can actually see the connections you are looking for.
The Metaphor: Imagine you are looking for specific lines (edges) connecting dots on a map. Some lines are thick and obvious; others are faint and hard to see.

This strategy asks: "How many people do I need to survey so that I can confidently say 'Yes, this line exists' or 'No, this line is empty'?"
They focus on the weakest link in your network. If you can detect the faintest connection you care about, you can definitely detect the stronger ones.
This ensures your study has enough "power" to find the truth, rather than just guessing.

The Takeaway

The paper provides a toolkit (and a free software package called designbgm) that helps researchers:

Count exactly how much their prior knowledge is worth in terms of sample size.
Plan their study size so that their new data is strong enough to dominate old beliefs.
Ensure they collect enough data to actually detect the connections they are interested in.

In short, it stops researchers from guessing how big their study needs to be. Instead, they can use a mathematical ruler to say, "My old study is worth 50 people, so to be sure my new data wins, I need to recruit at least 130 people." This makes the science more reliable and the studies more efficient.

Technical Summary: Prior Worth? Effective Sample Size and Sample Size Planning for Gaussian Graphical Models

Problem Statement
In Bayesian analysis of Gaussian Graphical Models (GGMs), researchers often utilize informative priors derived from previous studies to regularize the estimation of the precision matrix $\Theta$ . The standard conjugate priors for GGMs are the Wishart distribution (for complete graphs) and the G-Wishart distribution (for sparse graphs). These priors are governed by hyperparameters: degrees of freedom ( $\nu$ ) and a scale matrix ( $\Psi$ ).

A critical gap exists in the literature regarding the quantification of the "prior effective sample size" (ESS) for these multivariate, constrained distributions. While $\nu$ is often interpreted as the size of the prior study, it does not accurately reflect the actual information content of the prior in units of independent observations. This is because the entries of the precision matrix are dependent, and the informativeness of the prior depends on the full covariance structure encoded in $\Psi$ and the graph topology $G$ . Without a principled measure of prior ESS, researchers lack a basis for sample size planning: they cannot determine how large a new dataset must be for the data to dominate the prior, nor can they effectively design studies using Bayes Factor Design Analysis (BFDA) when informative priors are involved. Existing ESS methods are largely restricted to univariate or unconstrained multivariate settings and do not extend to the Wishart or G-Wishart distributions.

Methodology
The authors formalize a pre-data ESS framework for GGMs, operating exclusively before data collection to ensure applicability to study design. The methodology involves three main components:

Adaptation of ESS Estimators: Five pre-data ESS estimators, originally developed for univariate or general multivariate settings (Neuenschwander et al., 2020), are adapted to the precision matrix $\Theta$ under Wishart and G-Wishart priors. These include:
- Variance Ratio (VR)
- Precision Ratio (PR)
- Morita-Thall-Müller (MTM)
- Pennello-Thompson (PT)
- Expected Local Information Ratio (ELIR)
The authors define two aggregation schemes for these estimators:
- Global ESS: Computed via the determinant ratio of the numerator and denominator matrices (representing expected information and prior variance, respectively), raised to the power of $1/d$ (where $d = p(p+1)/2$ ).
- Parameterwise ESS: Derived via Cholesky decomposition of the numerator and denominator matrices, providing an ESS value for each unique entry $\theta_{ij}$ of the precision matrix.
For the Wishart prior, analytical closed-form expressions are derived. For the G-Wishart prior, where normalizing constants are intractable for non-decomposable graphs, the estimators are computed via Monte Carlo simulation.
Data-to-Prior Information Ratio (DPIR): A novel planning strategy is introduced to determine the sample size $n^\star$ required for the data to dominate the prior. The DPIR is defined as the ratio of the determinant of the expected Fisher information of the data to the determinant of the prior's observed Fisher information. The optimal sample size is the minimum $n$ such that the probability of the data information exceeding the prior information (by a factor $\xi \ge 1$ ) reaches a target threshold $\tau$ (default 0.99) under the prior predictive distribution.
Bayes Factor Design Analysis (BFDA) Extension: The authors extend BFDA to GGMs to determine the sample size required to achieve sufficient evidence for edge inclusion or exclusion. This involves:
- Formulating hypotheses for conditional independence ( $H_0: \rho_{ij} = 0$ vs. $H_1: \rho_{ij} \neq 0$ ).
- Using the Savage-Dickey density ratio for complete graphs and Monte Carlo estimation of normalizing constants for sparse graphs.
- Selecting a "planning edge" based on the $q$ -th quantile of prior partial correlations to ensure the sample size is sufficient for the weakest detectable edge, leveraging the monotonicity of required sample size with respect to effect size.

Key Results
Two simulation studies were conducted to validate the proposed methods:

Simulation Study 1 (ESS Behavior):
- The MTM, PT, and ELIR estimators were found to depend only on $\nu$ and the number of variables $p$ , rendering them invariant to the scale matrix $\Psi$ and graph structure.
- The VR and PR estimators were found to be sensitive to the full structure of $\Psi$ and the graph topology.
- A systematic "Jensen gap" was observed between VR and PR. The VR estimator generally yields values greater than $\nu$ ( $ESS_{VR} > \nu$ ), while the PR estimator yields values less than $\nu$ ( $ESS_{PR} < \nu$ ).
- The gap between VR and PR increases with network size ( $p$ ) and decreases with prior study size ( $\nu$ ).
- For the G-Wishart prior, the ratio $ESS/\nu$ exhibits non-monotonic behavior with respect to network density. $ESS_{VR}/\nu$ peaks at intermediate densities, while $ESS_{PR}/\nu$ decreases monotonically as density increases. This suggests that denser networks reduce the effective information of the prior relative to $\nu$ due to increased parameter dependencies.
Simulation Study 2 (Sample Size Planning):
- The DPIR and BFDA strategies were validated across various graph structures (Erdős-Rényi, Barabási-Albert, Watts-Strogatz) and prior specifications.
- The results confirmed that the two strategies target complementary goals: DPIR ensures data dominance over the prior, while BFDA ensures sufficient power for specific edge detection.
- The planned sample sizes ( $n^\star$ ) were shown to be sensitive to prior misspecification, particularly regarding the choice of the planning edge and the strength of the prior signal.
- The study demonstrated that relying solely on $\nu$ for sample size planning can be misleading; the actual ESS and the resulting $n^\star$ can differ significantly depending on the graph density and the specific geometry of the prior precision matrix.

Significance and Contributions
The paper claims to fill a critical gap in Bayesian GGM literature by providing the first principled framework for quantifying prior ESS and conducting sample size planning for these models. The specific contributions are:

Formalization of Pre-Data ESS: Deriving global and parameterwise ESS measures for Wishart and G-Wishart priors, acknowledging the dependence structure inherent in precision matrices.
Planning Strategies: Introducing the DPIR procedure and extending BFDA to GGMs, offering two complementary routes for determining sample size that account for prior informativeness.
Software Implementation: Providing an open-source R package, designbgm, which implements all estimators and the full DPIR–BFDA pipeline, making these methods accessible to researchers.

The authors conclude that without these tools, researchers eliciting informative priors for GGMs lack a rigorous basis for determining sample sizes, potentially leading to studies where the prior dominates the posterior or where edge detection is underpowered. The proposed methods allow researchers to assess whether their planned study carries sufficient information to outweigh the prior and to design studies that achieve conclusive evidence for specific network edges.