Pith. sign in

REVIEW 44 references

This survey is the first to organize uncertainty quantification methods for symbolic regression into frequentist, Bayesian, and model selection directions.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

A survey claiming to be the first comprehensive review of uncertainty quantification in symbolic regression, organized into three research directions and noting the area remains underexplored.

T0 review reviewed 2026-06-28 challenge →

load-bearing objection First dedicated survey on UQ in symbolic regression that splits the literature into frequentist, Bayesian, and model selection buckets; useful map of a thin area but stays at the level of organization rather than new analysis.

arxiv 2606.06567 v1 pith:BOCMVVJH submitted 2026-06-04 cs.LG

Are you sure? A Comprehensive and Comprehensible Survey of Uncertainty Quantification in Symbolic Regression

Julia Reuter , Fabricio Olivetti de Franca This is my paper
classification cs.LG
keywords symbolic regressionuncertainty quantificationfrequentist methodsBayesian methodsmodel selectionsurvey
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The reading

Symbolic regression discovers mathematical expressions from data but has limited support for uncertainty quantification, restricting its use in real-world decisions. The paper introduces core UQ concepts and reviews existing methods, grouping them into frequentist, Bayesian, and model selection approaches. A sympathetic reader would care because UQ informs model reliability, helps account for data noise to reduce overfitting, and supports informed decision-making. The survey finds UQ in SR remains underexplored and calls for more work on reliable methods.

Core claim

This survey is the first to clearly address this issue, with the objective of introducing essential UQ concepts and reviewing the current literature on UQ in SR, which can be broadly organized into three research directions: frequentist, Bayesian, and model selection. Despite its importance, UQ in SR is still underexplored, which motivates further research into reliable UQ methods for SR.

What carries the argument

The classification of UQ methods in symbolic regression into frequentist, Bayesian, and model selection research directions.

Load-bearing premise

The existing literature on UQ in SR is sufficiently developed and classifiable into the three stated directions without significant omissions or alternative organizing frameworks that would change the survey's conclusions.

What would settle it

A substantial body of UQ methods in symbolic regression that cannot be placed into frequentist, Bayesian, or model selection categories would challenge the survey's organizational framework.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • UQ provides information about model reliability in symbolic regression.
  • Accounting for uncertainty in the data can help avoid overfitting.
  • UQ offers insights that support decision-making processes.
  • The underexplored state of UQ in SR motivates development of new reliable methods.

Where Pith is reading between the lines

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

  • The three-direction framework could help practitioners select appropriate UQ techniques when applying symbolic regression.
  • If new hybrid methods appear, they might require updating or expanding the classification.
  • The survey could serve as a foundation for creating benchmarks that compare UQ performance across SR algorithms.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Circularity Check

0 steps flagged

No significant circularity

full rationale

This is a literature survey whose central claims consist of (a) introducing UQ concepts and (b) organizing existing external papers into the three stated directions (frequentist, Bayesian, model selection) while noting the area is underexplored. No equations, fitted parameters, derivations, or self-referential definitions appear in the provided text. The classification is presented as a broad review of outside work rather than a result derived from the authors' own prior results or ansatzes. Self-citations, if present, are not load-bearing for the organizational claim and do not reduce any prediction to an input by construction. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The survey rests on standard domain assumptions from statistics and machine learning; no free parameters, new entities, or ad-hoc axioms are introduced by the paper itself.

axioms (1)
  • domain assumption UQ provides important information about the model reliability, which can both help to avoid overfitting by accounting for uncertainty in the data, and provide insights for decision-making.
    Invoked in the abstract as background motivation for the survey.

reviewed 2026-06-28 · how reviews work

0 comments
Cite this review

Pith. "Pith review of Are you sure? A Comprehensive and Comprehensible Survey of Uncertainty Quantification in Symbolic Regression." pith.science (2026). https://pith.science/paper/BOCMVVJH

@misc{pith2026260606567,
  author       = {Pith},
  title        = {Pith review of: Are you sure? A Comprehensive and Comprehensible Survey of Uncertainty Quantification in Symbolic Regression},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BOCMVVJH}},
  note         = {Machine review of arXiv:2606.06567}
}
Share X Bluesky LinkedIn Reddit HN
read the original abstract

Symbolic regression (SR) is a class of methods that systematically explore the space of mathematical functions to discover models that accurately capture the underlying relationships in a dataset. Despite recent advances in the field, a lack of support for uncertainty quantification (UQ) limits its adoption in real-world decision processes. In regression analysis, UQ provides important information about the model reliability, which can both help to avoid overfitting by accounting for uncertainty in the data, and provide insights for decision-making. This survey is the first to clearly address this issue, with the objective of introducing essential UQ concepts and reviewing the current literature on UQ in SR, which can be broadly organized into three research directions: frequentist, Bayesian, and model selection. Despite its importance, UQ in SR is still underexplored, which motivates further research into reliable UQ methods for SR.

Figures

Figures reproduced from arXiv: 2606.06567 by Fabricio Olivetti de Franca, Julia Reuter.

Figure 1
Figure 1. Figure 1: Illustration of a multimodal likelihood search space with two [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Credible interval for 1 − α = 0.9. The lower and upper bounds is defined by the darker shaded region containing the 90% of the mass. It is a region of the parameter space that contains (1−α)100% of the posterior probability mass (see [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of prior (dashed blue line), likelihood (solid black line), [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Example of a confidence region of θ0 and θ1 using profile likelihood as introduced by [36]. This plot shows that the interval for θ1 is wider for smaller values of θ0 and tighter for larger values. regression models with SR, the authors use the pinball loss as the fitness function: Lτ (yi , fi) = ( (τ − 1)(yi − fi) if yi − fi ≥ 0, τ (yi − fi) if yi − fi < 0 . The quantile regression minimizes the effective… view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

44 extracted references · 4 canonical work pages · 2 internal anchors

  1. [1]

    J. R. Koza,Genetic Programming: On the Programming of Computers by Means of Natural Selection. Cambridge, MA, USA: MIT Press, 1992

    1992

  2. [2]

    Kronberger, B

    G. Kronberger, B. Burlacu, M. Kommenda, S. M. Winkler, and M. Af- fenzeller,Symbolic Regression. Chapman & Hall / CRC Press, 2024

    2024

  3. [3]

    Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods,

    E. H ¨ullermeier and W. Waegeman, “Aleatoric and epistemic uncertainty in machine learning: an introduction to concepts and methods,”Machine Learning, vol. 110, no. 3, pp. 457–506, Mar. 2021

    2021

  4. [4]

    Pawitan,In all likelihood: statistical modelling and inference using likelihood, paperback ed

    Y . Pawitan,In all likelihood: statistical modelling and inference using likelihood, paperback ed. Oxford: Oxford University Press, 2001

    2001

  5. [5]

    D. M. Bates and D. G. Watts,Nonlinear Regression Analysis and Its Applications, 1st ed., ser. Wiley Series in Probability and Statistics. Wiley, Aug. 1988

    1988

  6. [6]

    Variable selection for joint mean and dispersion models of the inverse Gaussian distribution,

    L. Wu and H. Li, “Variable selection for joint mean and dispersion models of the inverse Gaussian distribution,”Metrika, vol. 75, no. 6, pp. 795–808, Aug. 2012

    2012

  7. [7]

    K. P. Murphy,Probabilistic Machine Learning: An introduction. MIT Press, 2022

    2022

  8. [8]

    Nocedal and S

    J. Nocedal and S. J. Wright,Numerical optimization, ser. Springer series in operations research. New York: Springer, 1999

    1999

  9. [9]

    The Limiting Distributions of Certain Statistics,

    J. L. Doob, “The Limiting Distributions of Certain Statistics,”The Annals of Mathematical Statistics, vol. 6, no. 3, pp. 160–169, Sep. 1935

    1935

  10. [10]

    G. A. F. Seber and C. J. Wild,Nonlinear Regression, 1st ed., ser. Wiley Series in Probability and Statistics. Wiley, Feb. 1989

    1989

  11. [11]

    Conformal Prediction: A Gentle Introduction,

    A. N. Angelopoulos and S. Bates, “Conformal Prediction: A Gentle Introduction,”Foundations and Trends in Machine Learning, vol. 16, no. 4, pp. 494–591, Mar. 2023

    2023

  12. [12]

    Gelman, J

    A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari, and D. B. Rubin,Bayesian Data Analysis, 3rd ed., ser. Texts in Statistical Science Series. CHAPMAN & HALL/CRC Press, 2013

    2013

  13. [13]

    MCMC algorithms for constrained variance matrices,

    W. J. Browne, “MCMC algorithms for constrained variance matrices,” Computational Statistics & Data Analysis, vol. 50, no. 7, pp. 1655–1677, Apr. 2006

    2006

  14. [14]

    Adaptive Rejection Metropolis Sampling within Gibbs Sampling,

    W. R. Gilks, N. G. Best, and K. K. C. Tan, “Adaptive Rejection Metropolis Sampling within Gibbs Sampling,”Applied Statistics, vol. 44, no. 4, p. 455, 1995

    1995

  15. [15]

    Fractional Bayes Factors for Model Comparison,

    A. O’Hagan, “Fractional Bayes Factors for Model Comparison,”Journal of the Royal Statistical Society Series B: Statistical Methodology, vol. 57, no. 1, pp. 99–118, Jan. 1995

    1995

  16. [16]

    Model Selection and the Principle of Minimum Description Length,

    M. H. Hansen and B. Yu, “Model Selection and the Principle of Minimum Description Length,”Journal of the American Statistical Association, vol. 96, no. 454, pp. 746–774, Jun. 2001

    2001

  17. [17]

    Modeling by shortest data description,

    J. Rissanen, “Modeling by shortest data description,”Automatica, vol. 14, no. 5, pp. 465–471, Sep. 1978

    1978

  18. [18]

    Bayesian model selection for reducing bloat and overfitting in genetic programming for symbolic regression,

    G. F. Bomarito, P. E. Leser, N. C. M. Strauss, K. M. Garbrecht, and J. D. Hochhalter, “Bayesian model selection for reducing bloat and overfitting in genetic programming for symbolic regression,” inProc. of the Genetic and Evolutionary Computation Conf. Companion. Boston Massachusetts: ACM, Jul. 2022, pp. 526–529

    2022

  19. [19]

    Au- tomated learning of interpretable models with quantified uncertainty,

    G. Bomarito, P. Leser, N. Strauss, K. Garbrecht, and J. Hochhalter, “Au- tomated learning of interpretable models with quantified uncertainty,” Computer Methods in Applied Mechanics and Engineering, vol. 403, p. 115732, Jan. 2023

    2023

  20. [20]

    Comparing Methods for Estimating Marginal Likelihood in Symbolic Regression,

    P. Leser, G. Bomarito, G. Kronberger, and F. Olivetti De Franc ¸a, “Comparing Methods for Estimating Marginal Likelihood in Symbolic Regression,” inProc. of the Genetic and Evolutionary Computation Conf. Companion. Melbourne VIC Australia: ACM, Jul. 2024, pp. 2058–2066

    2024

  21. [21]

    Bayesian symbolic regression via posterior sampling,

    G. Bomarito and P. Leser, “Bayesian symbolic regression via posterior sampling,”Philosophical Transactions of the Royal Society A: Math- ematical, Physical and Engineering Sciences, vol. 384, no. 2317, p. 20240590, Apr. 2026

    2026

  22. [22]

    Priors for symbolic re- gression,

    D. J. Bartlett, H. Desmond, and P. G. Ferreira, “Priors for symbolic re- gression,” inProc. of the Companion Conf. on Genetic and Evolutionary Computation, Jul. 2023, pp. 2402–2411

    2023

  23. [23]

    A Bayesian machine scientist to aid in the solution of challenging scientific problems,

    R. Guimer `a, I. Reichardt, A. Aguilar-Mogas, F. A. Massucci, M. Mi- randa, J. Pallar`es, and M. Sales-Pardo, “A Bayesian machine scientist to aid in the solution of challenging scientific problems,”Science Advances, vol. 6, no. 5, p. eaav6971, Jan. 2020

    2020

  24. [24]

    Bayesian symbolic regression: auto- mated equation discovery from a physicist’s perspective,

    R. Guimer `a and M. Sales-Pardo, “Bayesian symbolic regression: auto- mated equation discovery from a physicist’s perspective,”Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engi- neering Sciences, vol. 384, no. 2317, p. 20250089, Apr. 2026

    2026

  25. [25]

    Estimation of probabilities from sparse data for the language model component of a speech recognizer,

    S. Katz, “Estimation of probabilities from sparse data for the language model component of a speech recognizer,”IEEE Trans. Acoust., Speech, Signal Process., vol. 35, no. 3, pp. 400–401, Mar. 1987

    1987

  26. [26]

    Exhaustive Symbolic Regression,

    D. J. Bartlett, H. Desmond, and P. G. Ferreira, “Exhaustive Symbolic Regression,”IEEE Trans. Evol. Computat., vol. 28, no. 4, pp. 950–964, Aug. 2024

    2024

  27. [27]

    Marginalised Normal Regression: Un- biased curve fitting in the presence of x-errors,

    D. Bartlett and H. Desmond, “Marginalised Normal Regression: Un- biased curve fitting in the presence of x-errors,”The Open Journal of Astrophysics, vol. 6, p. 10.21105/astro.2309.00948, Nov. 2023. 18

    work page doi:10.21105/astro.2309.00948 2023

  28. [28]

    The Inefficiency of Genetic Programming for Symbolic Regression,

    G. Kronberger, F. Olivetti De Franca, H. Desmond, D. J. Bartlett, and L. Kammerer, “The Inefficiency of Genetic Programming for Symbolic Regression,” inParallel Problem Solving from Nature – PPSN XVIII, M. Affenzeller, S. M. Winkler, A. V . Kononova, H. Trautmann, T. Tuˇsar, P. Machado, and T. B ¨ack, Eds. Cham: Springer Nature Switzerland, 2024, vol. 151...

    2024

  29. [29]

    Ensemble Bayesian Model Averaging in Genetic Programming,

    A. Agapitos, M. O’Neill, and A. Brabazon, “Ensemble Bayesian Model Averaging in Genetic Programming,” in2014 IEEE Congr. on Evolu- tionary Computation. Beijing, China: IEEE, Jul. 2014, pp. 2451–2458

    2014

  30. [30]

    Hierarchical Bayesian Operator-induced Symbolic Regression Trees for Structural Learning of Scientific Expressions,

    S. Roy, P. Dey, D. Pati, and B. K. Mallick, “Hierarchical Bayesian Operator-induced Symbolic Regression Trees for Structural Learning of Scientific Expressions,” 2025, https://arxiv.org/abs/2509.19710

    work page internal anchor Pith review arXiv 2025

  31. [31]

    VaSST: Variational Infer- ence for Symbolic Regression using Soft Symbolic Trees,

    S. Roy, P. Dey, and B. K. Mallick, “VaSST: Variational Infer- ence for Symbolic Regression using Soft Symbolic Trees,” 2026, https://arxiv.org/abs/2602.23561

    work page internal anchor Pith review arXiv 2026

  32. [32]

    An Introduction to Variational Autoen- coders,

    D. P. Kingma and M. Welling, “An Introduction to Variational Autoen- coders,”Foundations and Trends® in Machine Learning, vol. 12, no. 4, pp. 307–392, Nov. 2019

    2019

  33. [33]

    K. P. Murphy,Probabilistic Machine Learning: Advanced Topics. MIT Press, 2023

    2023

  34. [34]

    Uncertainty quantification based on symbolic regression and probabilistic programming and its application,

    Y . Zhao and H. Zhao, “Uncertainty quantification based on symbolic regression and probabilistic programming and its application,”Machine Learning with Applications, vol. 20, p. 100632, Jun. 2025

    2025

  35. [35]

    Symbolic Quantile Regression for the Interpretable Prediction of Conditional Quantiles,

    C. O. Hoekstra and F. d. Hengst, “Symbolic Quantile Regression for the Interpretable Prediction of Conditional Quantiles,” 2025

    2025

  36. [36]

    Prediction Intervals and Confidence Regions for Symbolic Regression Models based on Likelihood Profiles,

    F. O. de Franca and G. Kronberger, “Prediction Intervals and Confidence Regions for Symbolic Regression Models based on Likelihood Profiles,” 2022, https://arxiv.org/abs/2209.06454

    work page arXiv 2022

  37. [37]

    Discovering Unmodeled Components in Astrodynamics with Symbolic Regression,

    M. Manzi and M. Vasile, “Discovering Unmodeled Components in Astrodynamics with Symbolic Regression,” in2020 IEEE Congress on Evolutionary Computation. Glasgow, UK: IEEE, Jul. 2020, pp. 1–7

    2020

  38. [38]

    Learning noise,

    M. D. Schmidt and H. Lipson, “Learning noise,” inProc. of the 9th annual conf. on Genetic and evolutionary computation. London England: ACM, Jul. 2007, pp. 1680–1685

    2007

  39. [39]

    Combining conformal prediction and genetic programming for symbolic interval regression,

    P. T. Thuong, N. X. Hoai, and X. Yao, “Combining conformal prediction and genetic programming for symbolic interval regression,” inProceed- ings of the Genetic and Evolutionary Computation Conference. Berlin Germany: ACM, Jul. 2017, pp. 1001–1008

    2017

  40. [40]

    Genetic programming using a minimum description length principle,

    H. Iba, H. de Garis, and S. Taisuke, “Genetic programming using a minimum description length principle,” inAdvances in Genetic Pro- gramming. Cambridge, MA, USA: MIT Press, Aug. 1994, vol. 1, pp. 265–284

    1994

  41. [41]

    Regularization approach to inductive genetic programming,

    N. Nikolaev and H. Iba, “Regularization approach to inductive genetic programming,”IEEE Trans. Evol. Computat., vol. 5, no. 4, pp. 359–375, Aug. 2001. 19 APPENDIXA SUPPLEMENTARYMATERIAL FOR BASIC CONCEPTS OFUQ A. Likelihood and Fisher information As introduced in Sec. II-A, the Jacobian is a matrix of partial derivatives of the model outputs with respect ...

    2001

  42. [42]

    By fitting the model into the training set and evaluating it into the validation, we get a more reliable unbiased estimation of the likelihood

    Cross validation:A straightforward approach to detect overfitting caused by aleatoric uncertainty iscross validation, where the data is randomly split into different sets for training and validation. By fitting the model into the training set and evaluating it into the validation, we get a more reliable unbiased estimation of the likelihood. The idea is t...

  43. [43]

    Akaike Information Criterion:TheAkaike Information Criterion(AIC) penalizes the model linearly by the number of parametersd[4, Sec. 3.5]. The codelength increases by 2for every additional unit of the log-likelihood or for every additional parameter AIC =−2ℓ N(ˆθ) + 2d. In this sense, this criterion emphasizes the minimization of the negative log-likelihoo...

  44. [44]

    The complexity penalty is scaled by the logarithm ofN, thus under a large sample regime, models with fewer parame- ters are preferred [7, Sec

    Bayesian Information Criterion:TheBayesian Infor- mation Criterion(BIC) penalizes the model by its number of parameters with a magnitude proportional to the number of samples BIC =−2ℓ N(ˆθ) +dlog(N). The complexity penalty is scaled by the logarithm ofN, thus under a large sample regime, models with fewer parame- ters are preferred [7, Sec. 5.2.5.1]. Comp...

This paper was first reviewed by grok-4.3 on June 28, 2026.