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Handling bounded response in high dimensions: a Horseshoe prior Bayesian Beta regression approach

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims to build the first Gibbs sampling algorithm for high-dimensional sparse Beta regression, using Polya-Gamma augmentation and a Horseshoe prior, together with the first posterior consistency and concentration-rate results…

desk verdict The paper's PG-augmented Gibbs sampler is built on a false identity, so it samples a different model, and the theory is an unproven sketch; the work should be rejected. read the letter →

arxiv 2505.22211 v1 pith:XSYP3RFS submitted 2025-05-28 stat.ME math.STstat.MLstat.TH

The Tien Mai This is my paper · ORCID
classification stat.MEmath.STstat.MLstat.TH MSC 62F1562J0762G20
keywords BetaregressionHorseshoepriorPolya-GammaaugmentationGibbssamplerhigh-dimensionalsparsityfractionalposteriorconcentrationboundedresponse
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 aims to make Bayesian Beta regression work in high dimensions, where the number of predictors exceeds the sample size. It proposes a fractional posterior combined with a Horseshoe prior, and claims the first Gibbs sampling algorithm for Beta regression by invoking Polya-Gamma augmentation to rewrite the log-likelihood in a quadratic form. It also claims the first theoretical guarantees for Bayesian Beta regression: posterior consistency and a concentration rate $\varepsilon_n = K s^*\log(p/s^*)/n$ in $\alpha$-Rényi divergence, with the bound remaining meaningful when $p$ grows faster than $n$. If these claims are right, proportion-type outcomes could be analyzed with a conjugate sampler in sparse settings where classical Beta regression cannot be fitted at all.

What carries the argument

The load-bearing object is the Polya-Gamma augmentation identity for a logistic kernel: $\frac{(e^{\eta})^{\varphi y}}{(1+e^{\eta})^{\varphi}} \propto e^{\kappa\eta}\int_0^\infty e^{-\omega\eta^2/2}p(\omega)\,d\omega$ with $\kappa = \varphi(y-\tfrac12)$. The paper applies this identity to the Beta regression likelihood to obtain a conditionally Gaussian form for each observation, which in turn yields a multivariate normal full conditional for $\beta$. The Horseshoe prior enters through its hierarchical inverse-gamma representation, giving conjugate updates for all shrinkage variances, and the fractional posterior $\pi_{n,\alpha}(\beta)\propto L_n(\beta)^\alpha\pi(\beta)$ supplies the theoretical framework in which the concentration proof is carried out.

What would settle it

Choose one observation with fixed $\varphi$, a grid of $\eta$ values, and a fixed $y\in(0,1)$; compute the true Beta likelihood $f(y;\mu(\eta),\varphi)$ and compare it with the marginal $\int \operatorname{PG}(\omega;\varphi,\eta)\exp(\kappa\eta-\omega\eta^2/2)\,d\omega$ over the grid. If the ratio of the two quantities changes with $\eta$, the augmentation does not reproduce the Beta likelihood and the sampler does not target the stated posterior.

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Extended reading notes

Core claim

The paper's central claim is that a Beta regression likelihood, written with $\mu_i = (1+e^{-\eta_i})^{-1}$ and precision $\varphi$, can be transformed into a Gaussian form in $\eta_i$ through the Polya-Gamma identity: $\log p(y_i \mid \eta_i) \propto \kappa_i\eta_i - \omega_i\eta_i^2/2$ with $\kappa_i = \varphi(y_i - 1/2)$. This makes the full conditional of the regression vector $\beta$ multivariate normal, so inference can proceed by a closed-form Gibbs sampler alternating between Polya-Gamma draws, Gaussian draws for $\beta$, and inverse-gamma draws for the Horseshoe scale parameters. The paper further claims that, with the tempered likelihood $L_n(\beta)^\alpha$ and the Horseshoe prior, the fractional posterior concentrates around $\beta_0$ at rate $\varepsilon_n = K s^*\log(p/s^*)/n$, measured in $\alpha$-Rényi divergence, Hellinger distance, and total variation; the posterior mean estimator satisfies the same rate in prediction loss $\|X^\top(\hat\beta-\beta_0)\|_2^2$.

Load-bearing premise

The sampler is exact only if each Beta observation's likelihood equals, up to a constant independent of the linear predictor, the logistic form $(e^{\eta})^{\varphi y}/(1+e^{\eta})^{\varphi}$; the actual Beta density depends on $\eta$ through $\mu=(1+e^{-\eta})^{-1}$ in the shape parameters and through the $\beta$ normalizing constant, so this equivalence is the premise on which the whole Gibbs scheme rests.

Editorial extensions

If this is right

  • Beta regression becomes usable in $p>n$ regimes: the method estimates coefficients and selects variables when the number of covariates exceeds the sample size, a setting where classical maximum-likelihood Beta regression cannot even be fitted.
  • If the sampler is exact, it replaces Metropolis-Hastings updates for Beta regression with closed-form conditional draws, making posterior inference for bounded responses substantially faster.
  • The claimed concentration bound says the fractional posterior learns the true coefficient vector at near-minimax rate $s^*\log(p/s^*)/n$ with probability at least $1-2/(n\varepsilon_n)$.
  • The numerical comparisons claim the Horseshoe-Beta method reduces estimation error and improves variable-selection precision relative to transformed Lasso in both low- and high-dimensional simulations.

Reading between the lines

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

  • If the Polya-Gamma rewrite is exact, the same augmentation strategy would plausibly extend to other unit-interval response models whose mean is linked through the logit function, not just the Beta family.
  • The proof structure, which bounds divergence by squared linear-predictor distance, suggests the concentration result may carry over to neighboring bounded-response families with smooth densities, conditional on a similar prior-mass lemma.
  • A natural testable extension is a zero-one inflated version of the sampler, where a point-mass mixture component is added while the Polya-Gamma augmentation handles the continuous part of the response.
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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 / 4 minor

Summary. The paper proposes a sparse Bayesian Beta regression model for high-dimensional bounded responses, using a fractional posterior with a Horseshoe prior. The authors claim two main contributions: a novel Gibbs sampling algorithm based on Polya-Gamma augmentation for posterior inference, and the first theoretical guarantees (posterior consistency and convergence rates) for Bayesian Beta regression. The manuscript includes simulation studies in low and high dimensions, a sensitivity analysis for the precision parameter, and a real-data application. The theoretical results are stated in Section 4 and rely on appendices for proofs.

Significance. The problem is practically relevant, and a correct sparse Bayesian Beta regression method with computable inference and theoretical guarantees would be a valuable addition. The paper, however, does not deliver these contributions as written. The central computational claim is based on an incorrect likelihood identity, and the theoretical results are not supported because the required lemmas are unproved and the appendix contains a placeholder instead of proofs. If the computational error were isolated, the method could be salvaged by using a different sampler, but that would change the paper's central novelty. The paper is therefore not publishable in its current form.

major comments (3)
  1. [Section 3] The Polya-Gamma identity used in Step 1, (e^{η})^{φ y}/(1+e^{η})^φ ∝ exp(κ_i η) ∫ exp(−ω η²/2)p(ω)dω with κ_i = φ(y_i − 1/2), is valid for a binomial logistic kernel but is not the Beta likelihood in Eq. (1). The Beta log-density contains the log-normalizer log Γ(φ) − log Γ(μ_i φ) − log Γ((1−μ_i)φ) and the terms (μ_i φ −1) log y_i + ((1−μ_i)φ−1) log(1−y_i), where μ_i depends on η_i through the logit link. The quadratic augmentation would require the log-density to be linear-quadratic in η_i, which it is not; for instance, the second derivative of the log-density with respect to η_i is not a constant independent of η_i. Consequently, the multivariate normal full conditional for β in Step 2 is not the full conditional under the stated posterior. The sampler targets a different, binomialized model, so the simulation results and trace plots cannot be interpreted as evidence about the Beta regression posterior the paper claims to fit.
  2. [Appendix A] The proofs of Lemmas 1, 2, and 3 are missing; the text reads 'Proofs of Lemma 1, 2, and 3 are given at the end of the proof section, ??', with a literal placeholder. These lemmas are the technical core needed to verify the KL conditions in Theorem 2 of [1] that underpin Theorem 1. Lemma 1 in particular must control the KL divergence between two Beta distributions with different means, which requires careful handling of the Beta normalizing constant; it is not proved, so the main concentration result is currently unsupported.
  3. [Section 4] Even granting Lemmas 1–3, the proof of Theorem 1 is not supplied. The paper cites Theorems 2 and 3 from Alquier and Ridgway [1] and Lemma 4 from the author's prior work [24], but it does not show that the required conditions hold for the Beta regression model. Specifically, it does not verify the existence of a distribution ρ_n satisfying ∫ KL(P_{β0}, P_β) ρ_n(dβ) ≤ ε_n and KL(ρ_n, π) ≤ n ε_n for the Beta likelihood with horseshoe prior, nor does it verify the second-moment condition needed for the high-probability bound in inequality (5). The unproved lemmas are not merely technical; they are exactly the steps that connect the prior mass bound of Lemma 4 to the KL divergence of the Beta model. The theoretical claims are therefore not established in the manuscript.
minor comments (4)
  1. [Throughout] The manuscript contains numerous typos and grammatical errors, including 'hierachical', 'conducte', 'P´olya' spacing, and inconsistent use of 'Cau+' for the half-Cauchy distribution. A careful copyedit is needed.
  2. [Table 1] In the n = 500, ρ_X = 0.5 block, the FDR for the Horseshoe method is reported as 0.99 (0.03), which appears to be an error; this value is inconsistent with the reported Precision and Specificity and should be corrected.
  3. [Section 5.1] The test set size is fixed at n_test = 30 regardless of training sample size; for n = 100 this is a sizable fraction, and for n = 1000 it is very small. Reporting results for a fixed test size makes comparisons across settings less informative.
  4. [Section 5.2] The claim that the Horseshoe method 'consistently' outperforms the Lasso is too strong for the s* = 20, n = 80 cases, where the Horseshoe recall is low (0.43 and 0.21) and the test prediction error is comparable to or worse than Lasso in some settings.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the convergence-rate theorem is a direct application of Alquier–Ridgway fed by an independent Horseshoe prior-mass lemma, and the sampler issue is a misapplied Pólya–Gamma identity, not a definitional reduction.

full rationale

The claimed derivation chain is not circular. The concentration theorem (Theorem 1) is obtained by instantiating the external fractional-posterior results of Alquier and Ridgway ([1], quoted as Theorems 2–3) with the prior-mass bound of Lemma 4. Lemma 4 is quoted from the author's own prior paper [24], a self-citation, but it is a parameter-free Horseshoe concentration statement whose stated assumptions (‖β0‖0 = s∗, ‖β0‖∞ ≤ C1) do not include the target Beta-regression posterior result, and the same class of result is externally available (e.g., [40]); under the standard here it is independent evidence, so it does not raise the circularity score. The Gibbs sampler's Gaussian full conditional rests on Step 2's assertion that 'we can rewrite the Beta log-likelihood ... as log p(yi|η) ∝ κiη − ωη²/2'. That assertion is false for the Beta likelihood displayed two lines earlier in Section 3 (the true log-density contains η-dependent gamma normalizing terms and η-dependent coefficients multiplying log yi and log(1−yi)); the equality does not hold, so the sampler targets a different, binomialized model. This is an error of validity, not circularity: the paper's own displayed Beta likelihood is not equal by construction to the binomial kernel, so the Gaussian conditional does not reduce to the stated posterior by definition—the identity is misapplied, not definitionally imposed. The theoretical derivation is nevertheless incomplete: Appendix A marks the proofs of Lemmas 1–3 with 'Proofs of Lemma 1, 2, and 3 are given at the end of the proof section, ??', so the KL conditions required by [1] are not actually verified in the manuscript. Incompleteness and mathematical error are distinct from circularity; no fitted parameter is renamed as a prediction and no rate is forced by a self-citation chain.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The main unsupported inputs are the KL-condition sieve whose existence is asserted, the borrowed Horseshoe mass bound from the author's prior work, the weak design assumption, and the unproved Beta divergence lemmas. No new physical or data entities are introduced.

free parameters (2)
  • phi = 10 in simulations; not estimated in the real application
    The Beta regression precision parameter is treated as known; the model assumes a fixed phi common to all observations. Section 5.3 shows results degrade when phi is misspecified, so the choice matters.
  • alpha = 0.99
    Fractional power for the tempered posterior, chosen by hand and fixed to 0.99 in all experiments; alpha < 1 is required for the theory.
assumptions (4)
  • ad hoc to paper There exists a distribution rho_n with integral KL(P_beta0, P_beta) rho_n(d beta) <= epsilon_n and KL(rho_n, pi) <= n epsilon_n for epsilon_n = K s* log(p/s*)/n.
    The proof of Theorem 1 invokes Theorem 2.6 of Alquier and Ridgway [1] but never constructs rho_n or verifies these conditions; the appendix contains no proof.
  • domain assumption pi_HS(norm(beta - beta0)_2 < delta) >= exp(-K s* log(p/s*)) for delta = sqrt(s* log(p/s*)/n).
    Lemma 4, cited from the author's earlier paper [24]; a standard Horseshoe prior concentration bound, but not reproved here and not shown to imply the KL conditions.
  • domain assumption Assumptions 1-4 are sufficient: p <= exp(n^b), norm(beta0)_inf <= C1, E norm(X)^2 <= Cx, and |X^T beta0| <= C2 a.s.
    These assumptions are standard in low dimension but the finite second moment on norm(X) in Assumption 3 is weak for high-dimensional random design; no proof shows it yields the stated rate.
  • domain assumption Lemmas 1-3: KL, Renyi, and log-likelihood Lipschitz bounds for Beta distributions with nearby means.
    These lemmas are stated without proof; Appendix A ends with a placeholder '??' instead of the promised proofs.

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Cite this review

Pith. "Pith review of Handling bounded response in high dimensions: a Horseshoe prior Bayesian Beta regression approach." pith.science (2026). https://pith.science/paper/XSYP3RFS

@misc{pith2026250522211,
  author       = {Pith},
  title        = {Pith review of: Handling bounded response in high dimensions: a Horseshoe prior Bayesian Beta regression approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XSYP3RFS}},
  note         = {Machine review of arXiv:2505.22211}
}
read the original abstract

Bounded continuous responses -- such as proportions -- arise frequently in diverse scientific fields including climatology, biostatistics, and finance. Beta regression is a widely adopted framework for modeling such data, due to the flexibility of the Beta distribution over the unit interval. While Bayesian extensions of Beta regression have shown promise, existing methods are limited to low-dimensional settings and lack theoretical guarantees. In this work, we propose a novel Bayesian approach for high-dimensional sparse Beta regression framework that employs a tempered posterior. Our method incorporates the Horseshoe prior for effective shrinkage and variable selection. Most notable, we propose a novel Gibbs sampling algorithm using P\'olya-Gamma augmentation for efficient inference in Beta regression model. We also provide the first theoretical results establishing posterior consistency and convergence rates for Bayesian Beta regression. Through extensive simulation studies in both low- and high-dimensional scenarios, we demonstrate that our approach outperforms existing alternatives, offering improved estimation accuracy and model interpretability. Our method is implemented in the R package ``betaregbayes" available on Github.

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Forward citations

Cited by 1 Pith paper

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Reference graph

Works this paper leans on

44 extracted references · 42 canonical work pages · cited by 1 Pith paper

  1. [1]

    and Ridgway, J

    Alquier, P. and Ridgway, J. (2020). Concentration of tempered posteriors and of their variational approximations. The Annals of Statistics , 48(3):1475– 1497

    work page 2020

  2. [24]

    Mai, T. T. (2024a). Concentration of a Sparse Bayesian Model With Horseshoe Prior in Estimating High-Dimensional Precision Matrix. Stat, 13(4):e70008

    work page 2024

  3. [2]

    Barndorff-Nielsen, O. E. and Jørgensen, B. (1991). Some para metric models on the simplex. Journal of multivariate analysis , 39(1):106–116

    work page 1991

  4. [3]

    L., Baz´ an, J

    Bayes, C. L., Baz´ an, J. L., and Garcıa, C. (2012). A new robust regression model for proportions. Bayesian Analysis , 7(4):841–866

    work page 2012

  5. [4]

    C., Lecu´ e, G., and Tsybakov, A

    Bellec, P. C., Lecu´ e, G., and Tsybakov, A. B. (2018). Slope meet s lasso: im- proved oracle bounds and optimality. The Annals of Statistics , 46(6B):3603– 3642

    work page 2018

  6. [5]

    G., and Willard, B

    Bhadra, A., Datta, J., Polson, N. G., and Willard, B. (2017). The Ho rse- shoe+ Estimator of Ultra-Sparse Signals. Bayesian Analysis , 12(4):1105– 1131

    work page 2017

  7. [6]

    Bhattacharya, A., Pati, D., and Yang, Y. (2019). Bayesian frac tional poste- riors. Annals of Statistics , 47(1):39–66

    work page 2019

  8. [7]

    G., Holmes, C

    Bissiri, P. G., Holmes, C. C., and Walker, S. G. (2016). A general fr amework for updating belief distributions. Journal of the Royal Statistical Society Series B: Statistical Methodology , 78(5):1103–1130

    work page 2016

Show all 44 references
  1. [8]

    M., Polson, N

    Carvalho, C. M., Polson, N. G., and Scott, J. G. (2010). The hors eshoe estimator for sparse signals. Biometrika, 97(2):465–480

    2010

  2. [9]

    Castillo, I., Schmidt-Hieber, J., and van der Vaart, A. (2015). Ba yesian linear regression with sparse priors. The Annals of Statistics , 43(5):1986–2018

    2015

  3. [10]

    and N´ u˜ nez-Ant´ on, V

    Cepeda-Cuervo, E. and N´ u˜ nez-Ant´ on, V. (2013). Spatia l double gener- alized beta regression models: Extensions and application to study q uality of education in colombia. Journal of Educational and Behavioral Statistics , 38(6):604–628

    2013

  4. [11]

    Chakraborty, A., Bhattacharya, A., and Mallick, B. K. (2020). Bayesian sparse multiple regression for simultaneous rank reduction and var iable selec- tion. Biometrika, 107(1):205–221

    2020

  5. [12]

    and Zeileis, A

    Cribari-Neto, F. and Zeileis, A. (2010). Beta regression in R. Journal of statistical software, 34:1–24

    2010

  6. [13]

    and Cribari-Neto, F

    Ferrari, S. and Cribari-Neto, F. (2004). Beta regression for modelling rates and proportions. Journal of applied statistics , 31(7):799–815

    2004

  7. [14]

    I., Arellano-Valle, R

    Figueroa-Z´ u˜ niga, J. I., Arellano-Valle, R. B., and Ferrari, S. L . (2013). Mixed beta regression: A bayesian perspective. Computational Statistics & Data Analysis , 61:137–147. 20

    2013

  8. [15]

    I., Bayes, C

    Figueroa-Z´ u˜ niga, J. I., Bayes, C. L., Leiva, V., and Liu, S. (20 22). Robust beta regression modeling with errors-in-variables: a bayesian appr oach and numerical applications. Statistical Papers, pages 1–24

  9. [16]

    A., and Calder ´ ın-Ojeda, E

    G´ omez-D´ eniz, E., Sordo, M. A., and Calder ´ ın-Ojeda, E. (2014). The log– lindley distribution as an alternative to the beta regression model wit h appli- cations in insurance. Insurance: mathematics and Economics , 54:49–57

    2014

  10. [17]

    and Ghosal, S

    Jeong, S. and Ghosal, S. (2021). Posterior contraction in spa rse generalized linear models. Biometrika, 108(2):367–379

    2021

  11. [18]

    Karlsson, P., M ˚ ansson, K., and Kibria, B. G. (2020). A liu estimat or for the beta regression model and its application to chemical data. Journal of Chemometrics, 34(10):e3300

    2020

  12. [19]

    and McCullough, B

    Kieschnick, R. and McCullough, B. D. (2003). Regression analys is of vari- ates observed on (0, 1): percentages, proportions and fractio ns. Statistical modelling, 3(3):193–213

    2003

  13. [20]

    Knoblauch, J., Jewson, J., and Damoulas, T. (2022). An optimiza tion- centric view on bayes’ rule: Reviewing and generalizing variational inf erence. Journal of Machine Learning Research , 23(132):1–109

    2022

  14. [21]

    Lemonte, A. J. and Baz´ an, J. L. (2016). New class of johnso n distributions and its associated regression model for rates and proportions. Biometrical Journal, 58(4):727–746

    2016

  15. [22]

    and Eugenio, E

    Liu, F. and Eugenio, E. C. (2018). A review and comparison of ba yesian and likelihood-based inferences in beta regression and zero-or-on e-inflated beta regression. Statistical methods in medical research , 27(4):1024–1044

    2018

  16. [23]

    Liu, J. (2024). Beta regression for double-bounded respons e with correlated high-dimensional covariates. Stat, 13(2):e663

    2024

  17. [25]

    Mai, T. T. (2024b). On high-dimensional classification by sparse general- ized bayesian logistic regression. arXiv preprint arXiv:2403.12832

    2024 arXiv

  18. [26]

    Mai, T. T. (2024c). On properties of fractional posterior in ge neralized reduced-rank regression. arXiv preprint arXiv:2404.17850

    2024 arXiv

  19. [27]

    Mai, T. T. (2025a). Concentration properties of fractional p osterior in 1-bit matrix completion. Machine Learning, 114(1):7

    2025

  20. [28]

    Mai, T. T. (2025b). High-dimensional Bayesian Tobit regression for cen- sored response with Horseshoe prior. arXiv preprint arXiv:2505.08288 . 21

    2025 arXiv

  21. [29]

    Mai, T. T. (2025c). Optimal sparse phase retrieval via a quasi- bayesian approach. arXiv preprint arXiv:2504.09509

    2025 arXiv

  22. [30]

    and Schmidt, D

    Makalic, E. and Schmidt, D. F. (2015). A simple sampler for the ho rseshoe estimator. IEEE Signal Processing Letters , 23(1):179–182

    2015

  23. [31]

    Martin, R., Mess, R., and Walker, S. G. (2017). Empirical bayes p os- terior concentration in sparse high-dimensional linear models. Bernoulli, 23(3):1822–1847

    2017

  24. [32]

    and Tang, Y

    Martin, R. and Tang, Y. (2020). Empirical priors for prediction in sparse high-dimensional linear regression. Journal of Machine Learning Research , 21(144):1–30

    2020

  25. [33]

    and Moineddin, R

    Meaney, C. and Moineddin, R. (2014). A monte carlo simulation st udy comparing linear regression, beta regression, variable-dispersion beta regres- sion and fractional logit regression at recovering average differen ce measures in a two sample design. BMC medical research metho...

    2014

  26. [34]

    M., and Ongaro, A

    Migliorati, S., Di Brisco, A. M., and Ongaro, A. (2018). A new regr ession model for bounded responses. Bayesian Analysis , 13(3):845–872

    2018

  27. [35]

    Mullen, R., Marshall, L., and McGlynn, B. (2013). A beta regressio n model for improved solar radiation predictions. Journal of applied meteorology and climatology, 52(8):1923–1938

    2013

  28. [36]

    Paolino, P. (2001). Maximum likelihood estimation of models with bet a- distributed dependent variables. Political Analysis, 9(4):325–346

    2001

  29. [37]

    and Vehtari, A

    Piironen, J. and Vehtari, A. (2017). Sparsity information and r egularization in the horseshoe and other shrinkage priors. Electronic Journal of Statistics , 11(2):5018–5051

    2017

  30. [38]

    G., Scott, J

    Polson, N. G., Scott, J. G., and Windle, J. (2013). Bayesian infer ence for logistic models using p´ olya–gamma latent variables.Journal of the American statistical Association, 108(504):1339–1349

    2013

  31. [39]

    and Shou, Y

    Smithson, M. and Shou, Y. (2017). Cdf-quantile distributions f or modelling random variables on the unit interval. British Journal of Mathematical and Statistical Psychology, 70(3):412–438

    2017

  32. [40]

    van der Pas, S., Szab´ o, B., and van der Vaart, A. (2017). Adaptive posterior contraction rates for the horseshoe. Electronic Journal of Statistics , 11:3196– 3225

    2017

  33. [41]

    and Harremos, P

    Van Erven, T. and Harremos, P. (2014). R´ enyi divergence a nd kullback- leibler divergence. IEEE Transactions on Information Theory , 60(7):3797– 3820. 22

    2014

  34. [42]

    and Notaro, A

    Vanni, L. and Notaro, A. (2025). The impact of study environm ent on student gpa ratios: A beta regression analysis. Research Gate, doi: 10.13140/RG.2.2.15437.14560

    2025

  35. [43]

    and Qiu, Z

    Zhang, P. and Qiu, Z. (2014). Regression analysis of proportio nal data using simplex distribution. Scientia Sinica Mathematica , 44:89

    2014

  36. [44]

    and Huang, X

    Zhou, H. and Huang, X. (2022). Bayesian beta regression for bounded responses with unknown supports. Computational Statistics & Data Analysis , 167:107345. 23

    2022

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