Group-R2 Decomposition Prior in Bayesian Regression

• The paper introduces a two-stage variance decomposition that allocates explained variance first across known predictor groups and then within each group.
• It defines a hierarchical shrinkage prior for linear regression by parameterizing overall signal strength through R², enabling distinct control over model fit and sparsity.
• Empirical results demonstrate improved predictive performance when signals are distributed across groups, with hyperparameter tuning guiding effective model complexity.

Searching arXiv for the named prior and closely related R2-based grouped shrinkage work. The Group-R2 decomposition prior is a hierarchical shrinkage prior for linear regression with known groups of predictors. Its defining construction is a prior on the model’s explained variance, $R^2$, followed by a two-stage decomposition of the induced total signal variance: first across groups, then within groups across individual coefficients. In this formulation, model complexity, sparsity, and group importance are parameterized through variance allocation rather than through coefficient magnitudes alone, giving an explicitly $R^2$-based grouped extension of the broader R2D2/R2-prior family (Aguilar et al., 16 Jul 2025).

1. Formal setting and problem class

The prior is developed for the linear regression model

$$y = b_0 J + \sum_{g=1}^G X_g b_g + \varepsilon, \qquad \varepsilon_i \sim \mathcal{N}(0,\sigma^2),$$

where predictors are partitioned into $G$ mutually exclusive groups and

$$b = (b_1',\dots,b_G')', \qquad b_g = (b_{g1},\dots,b_{gp_g})'.$$

Under the assumptions $\mathbb{E}(x)=0$, $\operatorname{Var}(x)=\Sigma_x$ with unit diagonal, and

$$\operatorname{Var}(b)=\sigma^2\Lambda,\qquad \Lambda=\mathrm{diag}(\lambda_1^2,\dots,\lambda_p^2),$$

the variance of the linear predictor is

$$\operatorname{Var}(x'b)=\sigma^2 \sum_{i=1}^p \lambda_i^2.$$

The total prior signal variance is denoted

$\tau^2 \coloneqq \sum_{i=1}^p \lambda_i^2,$

and the corresponding coefficient of determination is

$R^2$0

This places the prior in the class of R2-based priors, where the global signal level is specified through $R^2$1 rather than through a direct prior on a global scale parameter. The grouped extension retains the original one-to-one relationship between $R^2$2 and total signal variance, but adds a structured variance split that respects known predictor partitions (Aguilar et al., 16 Jul 2025).

2. Two-stage decomposition of explained variance

The central construction decomposes total prior variance in two nested stages.

At Stage I, variance is allocated across groups by a simplex vector

$R^2$3

so that group $R^2$4 receives

$R^2$5

At Stage II, each group-specific variance is allocated within the group by

$R^2$6

This yields coefficient-specific variances

$R^2$7

The same hierarchy induces a group-wise explained variance

$R^2$8

Hence the grouped construction makes group contributions directly interpretable as shares of total explained variance. If $R^2$9, then

$$y = b_0 J + \sum_{g=1}^G X_g b_g + \varepsilon, \qquad \varepsilon_i \sim \mathcal{N}(0,\sigma^2),$$0

and, conditionally on $$y = b_0 J + \sum_{g=1}^G X_g b_g + \varepsilon, \qquad \varepsilon_i \sim \mathcal{N}(0,\sigma^2),$$1,

$$y = b_0 J + \sum_{g=1}^G X_g b_g + \varepsilon, \qquad \varepsilon_i \sim \mathcal{N}(0,\sigma^2),$$2

with negative dependence across groups: $$y = b_0 J + \sum_{g=1}^G X_g b_g + \varepsilon, \qquad \varepsilon_i \sim \mathcal{N}(0,\sigma^2),$$3

This decomposition is the prior’s main structural innovation. It separates overall model fit, between-group allocation, and within-group allocation into distinct stochastic components, allowing sparsity to operate simultaneously at the group and coefficient levels (Aguilar et al., 16 Jul 2025).

3. Prior hierarchy and interpretation of hyperparameters

The complete prior takes the form

$$y = b_0 J + \sum_{g=1}^G X_g b_g + \varepsilon, \qquad \varepsilon_i \sim \mathcal{N}(0,\sigma^2),$$4

In the main development, $$y = b_0 J + \sum_{g=1}^G X_g b_g + \varepsilon, \qquad \varepsilon_i \sim \mathcal{N}(0,\sigma^2),$$5 and $$y = b_0 J + \sum_{g=1}^G X_g b_g + \varepsilon, \qquad \varepsilon_i \sim \mathcal{N}(0,\sigma^2),$$6 are often taken as symmetric Dirichlet distributions: $$y = b_0 J + \sum_{g=1}^G X_g b_g + \varepsilon, \qquad \varepsilon_i \sim \mathcal{N}(0,\sigma^2),$$7 The prior on explained variance is

$$y = b_0 J + \sum_{g=1}^G X_g b_g + \varepsilon, \qquad \varepsilon_i \sim \mathcal{N}(0,\sigma^2),$$8

with mean and precision

$$y = b_0 J + \sum_{g=1}^G X_g b_g + \varepsilon, \qquad \varepsilon_i \sim \mathcal{N}(0,\sigma^2),$$9

which induces

$G$0

The hyperparameters have distinct roles. The pair $G$1 governs the global prior on $G$2, hence overall signal strength and model complexity. The parameter $G$3 governs allocation across groups through $G$4, while $G$5 governs allocation within group $G$6 through $G$7. The paper emphasizes the following interpretations:

• If $G$8, the Dirichlet on $G$9 concentrates near simplex corners, so only a few groups get most of the signal.
• If $$b = (b_1',\dots,b_G')', \qquad b_g = (b_{g1},\dots,b_{gp_g})'.$$0, allocation across groups is roughly uniform.
• If $$b = (b_1',\dots,b_G')', \qquad b_g = (b_{g1},\dots,b_{gp_g})'.$$1, variance is spread more evenly across groups.

Likewise:

• If $$b = (b_1',\dots,b_G')', \qquad b_g = (b_{g1},\dots,b_{gp_g})'.$$2, within-group allocation is sparse, favoring only a few active coefficients in each group.
• If $$b = (b_1',\dots,b_G')', \qquad b_g = (b_{g1},\dots,b_{gp_g})'.$$3, allocation within the group is roughly uniform.
• Larger $$b = (b_1',\dots,b_G')', \qquad b_g = (b_{g1},\dots,b_{gp_g})'.$$4 makes within-group shrinkage weaker and signal more spread out.

A particularly useful coupling is

$$b = (b_1',\dots,b_G')', \qquad b_g = (b_{g1},\dots,b_{gp_g})'.$$5

which yields clean marginal distributions for coefficient variances and ensures that within-group shrinkage is typically stronger than group-level shrinkage (Aguilar et al., 16 Jul 2025).

4. Distributional properties and shrinkage behavior

Under the aligned choice $$b = (b_1',\dots,b_G')', \qquad b_g = (b_{g1},\dots,b_{gp_g})'.$$6, $$b = (b_1',\dots,b_G')', \qquad b_g = (b_{g1},\dots,b_{gp_g})'.$$7, $$b = (b_1',\dots,b_G')', \qquad b_g = (b_{g1},\dots,b_{gp_g})'.$$8, and $$b = (b_1',\dots,b_G')', \qquad b_g = (b_{g1},\dots,b_{gp_g})'.$$9 with $\mathbb{E}(x)=0$0, the paper proves two especially simple marginal results: $\mathbb{E}(x)=0$1 and

$\mathbb{E}(x)=0$2

After integrating out the local variance, each coefficient has marginal density

$\mathbb{E}(x)=0$3

where

$\mathbb{E}(x)=0$4

and $\mathbb{E}(x)=0$5 is the confluent hypergeometric function of the second kind.

Its near-zero behavior depends sharply on $\mathbb{E}(x)=0$6. As $\mathbb{E}(x)=0$7:

• If $\mathbb{E}(x)=0$8,

$\mathbb{E}(x)=0$9

so the density is unbounded at zero.

• If $\operatorname{Var}(x)=\Sigma_x$0,

$\operatorname{Var}(x)=\Sigma_x$1

again singular at zero.

• If $\operatorname{Var}(x)=\Sigma_x$2, the density is bounded and continuous at zero.

Its tail behavior is

$\operatorname{Var}(x)=\Sigma_x$3

Thus $\operatorname{Var}(x)=\Sigma_x$4 controls tail thickness; if $\operatorname{Var}(x)=\Sigma_x$5, the tails are heavier than Cauchy. The special case

$\operatorname{Var}(x)=\Sigma_x$6

recovers the horseshoe prior, for which the prior has bounded influence, meaning

$\operatorname{Var}(x)=\Sigma_x$7

The grouped hierarchy also admits a shrinkage-factor representation. In the normal means setting $\operatorname{Var}(x)=\Sigma_x$8,

$\operatorname{Var}(x)=\Sigma_x$9

and the posterior mean obeys

$$\operatorname{Var}(b)=\sigma^2\Lambda,\qquad \Lambda=\mathrm{diag}(\lambda_1^2,\dots,\lambda_p^2),$$0

The paper defines the effective number of nonzero coefficients as

$$\operatorname{Var}(b)=\sigma^2\Lambda,\qquad \Lambda=\mathrm{diag}(\lambda_1^2,\dots,\lambda_p^2),$$1

with groupwise version

$$\operatorname{Var}(b)=\sigma^2\Lambda,\qquad \Lambda=\mathrm{diag}(\lambda_1^2,\dots,\lambda_p^2),$$2

This supplies a direct prior-based measure of effective model complexity (Aguilar et al., 16 Jul 2025).

5. Relation to adjacent grouped $$\operatorname{Var}(b)=\sigma^2\Lambda,\qquad \Lambda=\mathrm{diag}(\lambda_1^2,\dots,\lambda_p^2),$$3-based priors

The Group-R2 decomposition prior belongs to a broader line of work extending $$\operatorname{Var}(b)=\sigma^2\Lambda,\qquad \Lambda=\mathrm{diag}(\lambda_1^2,\dots,\lambda_p^2),$$4-based shrinkage to structured variance allocation. A closely related formulation is the Group R2D2 (gR2D2) prior, which extends the original R2D2 prior to grouped variable selection by decomposing explained variance into group-level and within-group components (Yanchenko et al., 2024). In that construction,

$$\operatorname{Var}(b)=\sigma^2\Lambda,\qquad \Lambda=\mathrm{diag}(\lambda_1^2,\dots,\lambda_p^2),$$5

with

$$\operatorname{Var}(b)=\sigma^2\Lambda,\qquad \Lambda=\mathrm{diag}(\lambda_1^2,\dots,\lambda_p^2),$$6

The paper places a Dirichlet prior on

$$\operatorname{Var}(b)=\sigma^2\Lambda,\qquad \Lambda=\mathrm{diag}(\lambda_1^2,\dots,\lambda_p^2),$$7

and uses a double exponential (Laplace) coefficient prior with a scale-mixture representation. It also develops two within-group variants: gR2D2-D, with Dirichlet within-group allocation, and gR2D2-L, with logistic-normal within-group allocation. Relative to this formulation, the Group-R2 decomposition prior is best understood as a closely aligned two-stage $$\operatorname{Var}(b)=\sigma^2\Lambda,\qquad \Lambda=\mathrm{diag}(\lambda_1^2,\dots,\lambda_p^2),$$8-decomposition framework stated directly in terms of $$\operatorname{Var}(b)=\sigma^2\Lambda,\qquad \Lambda=\mathrm{diag}(\lambda_1^2,\dots,\lambda_p^2),$$9, $$\operatorname{Var}(x'b)=\sigma^2 \sum_{i=1}^p \lambda_i^2.$$0, and $$\operatorname{Var}(x'b)=\sigma^2 \sum_{i=1}^p \lambda_i^2.$$1.

A second adjacent development is the Generalized Decomposition R2 (GDR2) prior, which replaces the Dirichlet decomposition on the simplex with a logistic-normal prior to allow dependence structures beyond the negative dependence implied by the Dirichlet family (Aguilar et al., 2024). In that framework,

$$\operatorname{Var}(x'b)=\sigma^2 \sum_{i=1}^p \lambda_i^2.$$2

and dependence is specified on the log-ratio scale through the additive log-ratio transform. The paper states that one may assign the same weight to a group of variables by setting $$\operatorname{Var}(x'b)=\sigma^2 \sum_{i=1}^p \lambda_i^2.$$3 for $$\operatorname{Var}(x'b)=\sigma^2 \sum_{i=1}^p \lambda_i^2.$$4, and identifies incorporation of covariate grouping or covariate dependency information as a promising future direction. This suggests that the Group-R2 decomposition prior occupies the Dirichlet-centered end of a larger design space in which grouped variance decomposition can be made more flexible by replacing simplex priors rather than by changing the $$\operatorname{Var}(x'b)=\sigma^2 \sum_{i=1}^p \lambda_i^2.$$5-based hierarchy itself.

6. Empirical behavior and practical specification

Simulation results show that grouping helps most when signals are distributed across multiple predictors within groups or mixed/randomly assigned rather than concentrated in a single coefficient. In the lower-dimensional setting $$\operatorname{Var}(x'b)=\sigma^2 \sum_{i=1}^p \lambda_i^2.$$6, grouped priors improve predictive performance in distributed and mixed-signal scenarios, especially when $$\operatorname{Var}(x'b)=\sigma^2 \sum_{i=1}^p \lambda_i^2.$$7 is high, and parameter recovery improves broadly, particularly for zero coefficients. In the high-dimensional setting $$\operatorname{Var}(x'b)=\sigma^2 \sum_{i=1}^p \lambda_i^2.$$8, the benefits of grouping become stronger, especially for distributed signals. By contrast, when the truth is highly concentrated in a small number of coefficients within each group, grouped priors can over-shrink true nonzero coefficients and may perform worse than non-grouped priors (Aguilar et al., 16 Jul 2025).

The paper gives practical guidance for hyperparameter specification. It recommends:

• $$\operatorname{Var}(x'b)=\sigma^2 \sum_{i=1}^p \lambda_i^2.$$9 as a sensible default if heavy tails and robust treatment of large coefficients are desired.
• $\tau^2 \coloneqq \sum_{i=1}^p \lambda_i^2,$0 for a uniform prior on $\tau^2 \coloneqq \sum_{i=1}^p \lambda_i^2,$1.
• $\tau^2 \coloneqq \sum_{i=1}^p \lambda_i^2,$2 for a bathtub-shaped prior, favoring either very small or very large $\tau^2 \coloneqq \sum_{i=1}^p \lambda_i^2,$3.
• $\tau^2 \coloneqq \sum_{i=1}^p \lambda_i^2,$4 when only a few groups are expected to matter.
• $\tau^2 \coloneqq \sum_{i=1}^p \lambda_i^2,$5 when only a few coefficients within each group are expected to matter.
• $\tau^2 \coloneqq \sum_{i=1}^p \lambda_i^2,$6 and $\tau^2 \coloneqq \sum_{i=1}^p \lambda_i^2,$7 as a balanced default if group structure is known but sparsity pattern is not.

The paper further recommends calibrating hyperparameters by simulating the prior distribution of $\tau^2 \coloneqq \sum_{i=1}^p \lambda_i^2,$8 or $\tau^2 \coloneqq \sum_{i=1}^p \lambda_i^2,$9, using effective model size as the operative measure of complexity. The reported uniform prior configuration was found to be a robust default when little is known about sparsity structure (Aguilar et al., 16 Jul 2025).

7. Scope, assumptions, and disambiguation

The prior assumes that group structure is known in advance. It is designed for settings with specified groups of predictors, and the paper explicitly notes that if groups are misspecified, variance may be allocated poorly, hurting inference. It also does not model unknown within-group dependence; it uses group membership rather than latent group discovery. A limiting case is $R^2$00, where the grouped model reduces to the ordinary R2-based coefficient-level prior (Aguilar et al., 16 Jul 2025).

The term should also be distinguished from a separate use of $R^2$01 in multi-objective optimization. The paper “Computing the Integral R2 Indicator by Perspective Mapping and Box Decomposition” studies the continuous integral $R^2$02 indicator, showing that it can be rewritten as a weighted measure of complements of unions of anchored boxes in reciprocal objective space (Emmerich, 29 Jun 2026). That work provides a decomposition-based structural interpretation of integral $R^2$03, but it does not introduce a “Group-R2 Decomposition Prior” in a Bayesian or statistical-prior sense. The shared notation therefore masks two distinct literatures: one centered on $R^2$04-based shrinkage in grouped regression, the other on decomposition methods for the integral $R^2$05 indicator in Pareto optimization.

In the Bayesian regression sense, the Group-R2 decomposition prior is most precisely characterized as a two-stage variance decomposition prior on $R^2$06. Its distinguishing feature is not merely grouped shrinkage, but the fact that group relevance and within-group sparsity are both expressed as allocations of prior explained variance. This suggests an interpretation of grouped regularization in which the primary object is not the coefficient vector itself, but the structured partition of the model’s total explainable signal.