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REVIEW 3 major objections 5 minor 14 references

R2 priors for Grouped Variance Decomposition in High-dimensional Regression

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The Group-R2 prior splits prior explained variance across known predictor groups and within them, improving recovery for spread-out signals while concentrated signals stay difficult.

desk verdict Useful grouped extension of R2 priors, but the theory assumes one hyperparameter coupling and the simulations use another—the central empirical claim needs rework before it can be trusted. read the letter →

arxiv 2507.11833 v2 pith:UHPFX6YL submitted 2025-07-16 stat.ME stat.APstat.OT

Javier Enrique Aguilar , David Kohns , Aki Vehtari , Paul-Christian Bürkner This is my paper
classification stat.MEstat.APstat.OT MSC 62F1562J07
keywords priorspecificationshrinkagepriorsvariancedecompositionhigh-dimensionalregressioncoefficientofdeterminationgroupedpredictorsBayesianvariableselectionR2
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces a class of shrinkage priors for Bayesian linear regression in which the prior distribution on the coefficient of determination R2 is split into two nested allocations: first across known groups of predictors, then across coefficients within each group. This lets a user state prior beliefs about how much variance each group explains, while preserving the desirable behavior of R2-based shrinkage priors: heavy tails for large signals and a spike at zero for noise. Using simulations with p=100 and p=500, the authors argue that the grouped version consistently improves parameter recovery and predictive performance over the nongrouped R2D2 prior when the true signal is distributed across predictors or randomly mixed, with larger gains at high R2 and in higher dimensions. When the signal is concentrated within groups, grouping gives little benefit and can over-shrink the truly nonzero coefficients, so the method's value depends on the signal pattern. The paper also derives the marginal densities, tail behavior, and a horseshoe special case, and offers hyperparameter guidance through the prior distribution of the effective number of non-zero coefficients.

What carries the argument

The two-stage variance decomposition: R2 ~ Beta(a1,a2) induces $τ^{2}$ = R2/(1−R2) ~ BetaPrime(GaG, a2); Stage I draws group proportions ϕ ~ Dirichlet_s(aG) so group variance is $τg^{2}$ = ϕg $τ^{2}$; Stage II draws within-group proportions φg ~ Dirichlet_s(cg) so each coefficient has variance $λ^{2}$_gl $σ^{2}$ = φ_gl ϕ_g $τ^{2}$ $σ^{2}$. With the coupling cg = aG/pg, the marginal coefficient variance is BetaPrime(cg, a2), and the marginal density of bgl can be written in closed form using a confluent hypergeometric function U; this same expression yields the spike at zero (cg ≤ 1/2), the tail exponent (a2), and the horseshoe boundary (cg = a2 = 1/2). The shrinkage factor κbl = 1/(1+$λ^{2}$_gl) and the effective number of non-zero coefficients meff, along with its group-wise version meff,g, turn the hyperparameters into interpretable sparsity diagnostics.

What would settle it

Generate a p=500, n=200 data set with R2=0.8 and a distributed signal in one active group, but fit the prior with a deliberately permuted group assignment that spreads the active group across several low-concentration groups; if the grouped prior's RMSE and ELPD then fall below the nongrouped R2D2 baseline, the claim of consistent gains in distributed-signal settings would be refuted.

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Extended reading notes

Core claim

The central claim is that decomposing the prior variance of R2 in two stages—across groups and then within groups—creates a group-aware shrinkage prior that keeps the good properties of R2D2 priors and gains when signal structure matches the grouping. Formally, with a symmetric Dirichlet across groups and within groups, each coefficient's prior variance factorizes as φgl φg $τ^{2}$ $σ^{2}$, and setting cg = aG/pg yields closed-form marginal Beta Prime variances. The prior is shown to have polynomial tails with exponent 2a2+1, heavier than Cauchy for a2 < 1/2, an unbounded density at zero for cg ≤ 1/2, and to coincide with the horseshoe prior at cg = a2 = 1/2. Simulation comparisons against the nongrouped R2D2 prior show the grouped version improves predictive ELPD and posterior RMSE in distributed, random-distributed, and random-coefficient scenarios, especially in the p=500, R2=0.8 setting, while concentrated-signal scenarios show little or no gain.

Load-bearing premise

The whole method depends on the user's grouping being right; the paper assumes the groups are known and fixed, and the authors state that wrong groups can over-shrink true effects and inflate noise groups.

Editorial extensions

If this is right

  • Setting R2 ~ Beta(GaG, a2) and cg = aG/pg cuts the hyperparameter search to two numbers while keeping closed-form marginal variances.
  • Users can translate prior beliefs about total and group-level explained variance directly into shrinkage, rather than setting uninterpretable scale parameters.
  • In high-dimensional simulations (p=500, n=200) with distributed or randomly mixed signals, the grouped prior improves predictive performance and parameter recovery over nongrouped R2D2, with gains increasing in aG and at R2=0.8.
  • The prior inherits bounded influence: large coefficients remain nearly unshrunk, matching horseshoe behavior when cg=a2=1/2.
  • The group-wise effective number of non-zero coefficients provides a pre-experiment diagnostic for matching hyperparameters to expected sparsity.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because the gains come from sharing variance information across group members, a natural testable extension is to learn the groups or use soft or overlapping assignments; the paper notes group misspecification as a limitation, leaving this open.
  • The negative correlation among within-group shrinkage factors suggests the prior implicitly favors selecting a small number of representatives per group; this could be checked by examining posterior inclusion patterns in groups with cg < 1.
  • The same two-stage decomposition could be ported to time-series settings where lags form natural groups, using the group-level R2 to encode beliefs about which lag block matters.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper introduces the Group-R2 decomposition prior for linear regression with known predictor groups. The prior places a Beta distribution on R2, distributes total variance across groups via a symmetric Dirichlet, distributes each group's variance within the group via a second symmetric Dirichlet, and uses normal kernels for coefficients. The authors derive theoretical properties under the alignment condition c_g = a_G/p_g: marginal distributions of group and coefficient variances, marginal coefficient densities, tail behavior, a horseshoe special case, and shrinkage-factor properties. They give hyperparameter guidance based on the effective number of nonzero coefficients. Simulation studies compare grouped versus nongrouped R2D2 priors at p = 100 and p = 500 with n = 200 under concentrated, distributed, random-concentrated, random-distributed, and random-coefficient signal scenarios, reporting ELPD and RMSE differences. The central claim is that Group-R2 priors consistently improve parameter recovery when the signal is distributed across predictors or randomly mixed, and improve prediction especially at high R2.

Significance. The paper has real strengths: it provides an interpretable two-stage construction that extends R2-based priors to grouped predictors, it ships code and data, it includes machine-checkable derivations for the aligned case, and it explicitly acknowledges the reliance on pre-specified group structures. If the main empirical claim is supported under the conditions recommended by the theory, the Group-R2 prior would be a useful addition to the shrinkage-prior toolbox. However, the current simulations abandon the alignment condition on which the theoretical results rest, so the practical significance is conditional on additional work.

major comments (3)
  1. [Section 3.2.2 vs Section 2.4 / Equation (18)] The simulations fix c_g = 0.5 with a_G in {0.1, 0.5, 1}, while Propositions 2, 4, 5, and 8 and Equation (18) assume c_g = a_G/p_g. With p_g = 10, the aligned values would be 0.01, 0.05, and 0.1, none of which equals 0.5. For a_G = 0.1 and p_g = 10, the beta-function argument in Equation (18), a_G - p_g c_g, equals -4.9, so the closed-form marginal for the shrinkage factors is not valid in the simulated configuration. The paper nevertheless says in Section 3.2.2 that it adopts the marginal formulations of Section 2.4 to set hyperparameters. The central empirical claim is therefore not backed by the paper's own theory in the configurations actually run.
  2. [Section 2.6 vs Section 3.2.2] Section 2.6 recommends the coupled choice c_g = a_G/p_g as a coherent way to reduce the hyperparameter space and obtain closed-form marginals, but Section 3.2.2 departs from this coupling for all grouped models, including R2-c and R2-d, without explaining the departure. The fixed c_g = 0.5 grid is justified only heuristically through the meff figures. The authors should either run the main simulations under the recommended coupling or provide theoretical support for the grid actually used and clarify how the practical recommendations in Section 2.6 apply to that grid.
  3. [Section 4 (Discussion) vs Section 3.2.3] The Discussion claims that 'in both lower- and high-dimensional settings, Group R2 priors consistently improve parameter recovery when the signal is distributed across predictors or randomly mixed.' This overstates the reported results. For the high-dimensional Random Coefficients setting, Section 3.2.3 states that improvement occurs only for a_G = 1 and R2 = 0.8, with little or no improvement otherwise (Figures 10 and 13). The claim should be narrowed to the conditions actually supported by the experiments, or the experiments need to be extended to justify the stronger claim.
minor comments (5)
  1. [Section 5.1, Proof of Proposition 3] The proof states E[phi_gj phi_gk] = c_g^2 / (p_g^2 (c_g p_g + 1)) for j != k. The correct expression for a symmetric Dirichlet is c_g / (p_g (c_g p_g + 1)) = c_g p_g / (p_g^2 (c_g p_g + 1)). The main-text formula in Proposition 3 appears correct, but the proof needs to be fixed.
  2. [Equation (15) and Equation (18)] The shrinkage-factor densities are not correctly normalized. The Dirichlet normalizing constant should include Gamma(c_g p_g) / Gamma(c_g)^{p_g}, and the unconditional density in Equation (18) should also include the factor 1 / B(a_G, a_2) from the BetaPrime prior on tau_g^2. As written, both expressions are missing these constants.
  3. [Figure 5 and Figure 11 captions] The captions say 'for groups of size 10' but then state p_g = 20 for all groups. This is contradictory and should be corrected.
  4. [Section 5.1, Proof of Proposition 2] After applying Proposition 10, the proof writes tau_g^2 | xi ~ BetaPrime(a_G, xi), but the correct statement is tau_g^2 | xi ~ Gamma(a_G, xi). The subsequent line uses the Gamma form, so this appears to be a typo.
  5. [Section 2.1] The DAG is referred to as 'Figure 2.1' in the text, but the figure is numbered 'Figure 2'. The cross-reference should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Group-R2 prior's theoretical results are derived from its stated construction, and the simulation claims are evaluated against a nongrouped baseline with a priori hyperparameter choices.

full rationale

The paper's derivation chain is self-contained. The Group-R2 prior is explicitly constructed in Section 2.1 as a hierarchical product of a BetaPrime global variance and two symmetric Dirichlet decompositions; the marginal, tail, and shrinkage results (Propositions 2, 4-8) follow from standard gamma/beta and confluent hypergeometric calculations with external supporting references such as Zhang et al. (2020) and Carvalho et al. (2010), not from the empirical claims they later support. The simulation study compares grouped and nongrouped R2 priors under hyperparameters (aG, cg) chosen a priori rather than fitted to the data, so the Delta-ELPD and Delta-RMSE results are not predictions of fitted inputs. Self-citations to Aguilar and Buerkner (2023, 2025) and Kohns et al. (2025) are used only to justify the normal kernel and prior conventions; no load-bearing claim depends on an unverified self-cited uniqueness theorem or ansatz. The paper candidly flags in Section 4 that pre-specified group structures are a limitation, and the mismatch between the aligned theory setting cg=aG/pg and the simulated cg=0.5 is a substantive external-validity and robustness concern, not a circular reduction of the paper's conclusions to its inputs.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central construction rests on a handful of modeling choices: standardized predictors, known disjoint groups, zero-mean prior with diagonal covariance, and symmetric Dirichlet distributions at both decomposition levels. The theory requires an additional alignment c_g = a_G/p_g to obtain closed-form marginal Beta prime distributions; this condition is not satisfied in the simulation study, which fixes c_g = 0.5 while varying a_G. The free parameters are the prior hyperparameters (a1, a2, a_G, c_g), all chosen a priori rather than fitted. No new physical entities are introduced; the Group-R2 prior is a mathematical construct.

free parameters (4)
  • a_G = 0.1, 0.5, 1 in simulations
    Group-level symmetric Dirichlet concentration; controls allocation of variance across groups.
  • c_g = 0.5 (or 1 for R2-d) in simulations
    Within-group symmetric Dirichlet concentration; controls allocation of variance within groups.
  • a2 = 0.5 in default models
    Second shape parameter of the Beta prior on R2; set to induce heavy tails (Proposition 6).
  • a1 = G a_G in default models; 1 for uniform R2-u
    First shape parameter of the Beta prior on R2; coupled to a_G in the recommended specification.
assumptions (6)
  • domain assumption Predictors are standardized: E(x)=0 and Var(x)=Σ with unit diagonal.
    Stated in Section 2.1 as the basis for the relationship R² = τ²/(τ² + 1).
  • domain assumption Independent, mutually exclusive groups of predictors are known in advance.
    The entire prior construction assumes a fixed partition of p predictors into G groups (Section 2.1).
  • domain assumption The coefficient prior has zero mean and diagonal covariance Var(b) = σ²Λ.
    Needed to express total variance as the sum of per-coefficient variances in Section 2.1.
  • domain assumption Within-group and group-level proportions follow symmetric Dirichlet distributions.
    This is a modeling choice (Section 2.1) that enables the closed-form results.
  • ad hoc to paper The concentration parameters are aligned as c_g = a_G/p_g to obtain closed-form marginal beta-prime distributions.
    Stated before Proposition 2; this alignment is assumed throughout the theoretical sections but is not used in the simulation study.
  • standard math Standard analytic facts about Dirichlet, Beta prime, confluent hypergeometric U, and Watson's lemma.
    Used in the proofs of Propositions 1-8; these are established results from the cited literature.

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Cite this review

Pith. "Pith review of R2 priors for Grouped Variance Decomposition in High-dimensional Regression." pith.science (2026). https://pith.science/paper/UHPFX6YL

@misc{pith2026250711833,
  author       = {Pith},
  title        = {Pith review of: R2 priors for Grouped Variance Decomposition in High-dimensional Regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UHPFX6YL}},
  note         = {Machine review of arXiv:2507.11833}
}
read the original abstract

We introduce the Group-R2 decomposition prior, a hierarchical shrinkage prior that extends R2-based priors to structured regression settings with known groups of predictors. By decomposing the prior distribution of the coefficient of determination R2 in two stages, first across groups, then within groups, the prior enables interpretable control over model complexity and sparsity. We derive theoretical properties of the prior, including marginal distributions of coefficients, tail behavior, and connections to effective model complexity. Through simulation studies, we evaluate the conditions under which grouping improves predictive performance and parameter recovery compared to priors that do not account for groups. Our results provide practical guidance for prior specification and highlight both the strengths and limitations of incorporating grouping into R2-based shrinkage priors.

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Reference graph

Works this paper leans on

14 extracted references · 6 canonical work pages

  1. [1]

    Dependency-Aware Shrinkage Priors for High Dimensional Regression

    Aguilar, J. E. and B¨ urkner, P.-C. (2025). “Dependency-Aware Shrinkage Priors for High Dimensional Regression.” arXiv preprint arXiv:2505.10715 . 15 Aguilar, J. E. and B¨ urkner, P.-C. (2023). “Intuitive joint priors for Bayesian linear multilevel models: The R2D2M2 prior.” Electronic Journal of Statistics , 17(1): 1711 –

    work page Pith review arXiv 2025

  2. [30]

    Stan: A Probabilistic Programming Language

    URL https://doi.org/10.1214/23-BA1371 2, 6, 12, 17, 19, 22 Brooks, Gelman, S., and Jones, A. (2011). Handbook of Markov Chain Monte Carlo. Chapman and Hall/CRC, 1 edition. 17, 18 Carpenter, B., Gelman, A., Hoffman, M. D., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M., Guo, J., Li, P., and Riddell, A. (2017). “Stan: A Probabilistic Programming Langua...

    work page arXiv 2011

  3. [32]

    Flexible Bayesian MIDAS: time-variation, group-shrinkage and sparsity

    URL https://doi.org/10.1214/25-BA1512 2, 5 Kohns, D. and Potjagailo, G. (2025). “Flexible Bayesian MIDAS: time-variation, group-shrinkage and sparsity.” Journal of Business & Economic Statistics , 1–17. 1 Li, C. and Li, H. (2008). “Network-constrained regularization and variable selection for analysis of genomic data.” Bioinformatics, 24(9): 1175–1182. UR...

    doi:10.1214/25-ba1512 2025

  4. [34]

    Supple- ment to

    URL https://doi.org/10.1214/25-BA1524 2, 5, 9 Aguilar, J. E., Kohns, D., Vehtari, A., and B¨ urkner, P.-C. (2025). “Supple- ment to ”R2 priors for Grouped Variance Decomposition in High-dimensional Regression”.” 8, 14, 18 Armagan, A., Dunson, D. B., Lee, J., Bajwa, W. U., and Strawn, N. (2013). “Posterior consistency in linear models under shrinkage prior...

    work page arXiv 2025

  5. [110]

    Theoretical guarantees for the horseshoe and other global-local shrinkage priors

    URL https://doi.org/10.1214/088342304000000017 3 Neal, R. M. (2011). MCMC using Hamiltonian dynamics . URL http://arxiv.org/abs/1206.1901 17 Olver, F., of Standards, N. I., (U.S.), T., Lozier, D., Boisvert, R., and Clark, C. (2010a). NIST Handbook of Mathematical Functions Hardback and CD-ROM . Cambridge University Press. URL https://books.google.de/books...

    arXiv 2011

  6. [228]

    Bayesian Variable Selection and Estimation for Group Lasso

    URL https://doi.org/10.1214/12-SS102 18 Xu, X. and Ghosh, M. (2015). “Bayesian Variable Selection and Estimation for Group Lasso.” Bayesian Analysis , 10(4). URL https://projecteuclid.org/journals/bayesian-analysis/ volume-10/issue-4/Bayesian-Variable-Selection-and-Estimation-for-Group-Lasso/ 10.1214/14-BA929.full 2 Xu, Z., Schmidt, D. F., Makalic, E., Qi...

    arXiv 2015

  7. [534]

    rstanarm: Bayesian applied regression modeling via Stan

    URL https://doi.org/10.1214/06-BA117A 6 Gelman, A., Carlin, J., Stern, H., Dunson, D., Vehtari, A., and Rubin, D. (2013). Bayesian Data Analysis, Third Edition . Chapman & Hall/CRC Texts in Statistical Science. Taylor & Francis. 1 Goodrich, B., Gabry, J., Ali, I., and Brilleman, S. (2020). “rstanarm: Bayesian applied regression modeling via Stan.” URL htt...

    doi:10.1214/06-ba117a 2013

  8. [718]

    A survey of Bayesian predictive methods for model assessment, selection and comparison

    URL https://doi.org/10.1214/20-BA1221 18 Vehtari, A. and Ojanen, J. (2012). “A survey of Bayesian predictive methods for model assessment, selection and comparison.” Statistics Surveys , 6(none): 142 –

    doi:10.1214/20-ba1221 2012

Show all 14 references
  1. [1000]

    Uncertainty Quantification for the Horseshoe (with Discussion)

    URL https://doi.org/10.1214/16-EJS1130 12, 13 Pas, S. L. v. d., Szab´ o, B., and Vaart, A. v. d. (2017). “Uncertainty Quantification for the Horseshoe (with Discussion).” Bayesian Analysis , 12(4): 1221 –

    2017 doi

  2. [1274]

    Sparsity information and regularization in the horseshoe and other shrinkage priors

    URL https://doi.org/10.1214/17-BA1065 11 Piironen, J. and Vehtari, A. (2017). “Sparsity information and regularization in the horseshoe and other shrinkage priors.” Electronic Journal of Statistics , 11(2): 5018 –

    2017 doi

  3. [1767]

    Generalized Decomposition Priors on R2

    URL https://doi.org/10.1214/23-EJS2136 2, 5 — (2025). “Generalized Decomposition Priors on R2.” Bayesian Analysis , 1 –

    2025 doi

  4. [2101]

    On the Distribution of the Product of Independent Beta Random Variables — Applications

    URL https://doi.org/10.1214/12-AOS1029 13 Coelho, C. A. and Alberto, R. P. (2021). “On the Distribution of the Product of Independent Beta Random Variables — Applications.” In Ghosh, I., Bal- akrishnan, N., and Ng, H. K. T. (eds.), Advances in Statistics - Theory and Applicati...

    2021 doi

  5. [2618]

    Conditions for posterior contraction in the sparse normal means problem

    URL https://doi.org/10.1214/14-EJS962 11, 13 Pas, S. L. v. d., Salomond, J.-B., and Schmidt-Hieber, J. (2016). “Conditions for posterior contraction in the sparse normal means problem.” Electronic Journal of Statistics , 10(1): 976 –

    2016 doi

  6. [5051]

    Local Shrinkage Rules, L´ evy Processes and Regularized Regression

    URL https://doi.org/10.1214/17-EJS1337SI 12, 15 Polson, N. G. and Scott, J. G. (2012). “Local Shrinkage Rules, L´ evy Processes and Regularized Regression.” Journal of the Royal Statistical Society Series B: Statistical Methodology , 74(2): 287–311. URL https://doi.org/10.1111...

    2012

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