Pith. sign in

REVIEW 4 major objections 5 minor 30 references

Bayesian symbolic regression: Automated equation discovery from a physicists' perspective

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read Symbolic regression should rank candidate equations by Bayesian plausibility, and that ranking reduces to minimizing a single description-length score that unifies fit, parameter cost, and prior complexity.

desk verdict A clear, well-written Bayesian manifesto for symbolic regression whose practical criterion inherits the advertised guarantees only under approximations the paper doesn't verify. read the letter →

arxiv 2507.19540 v1 pith:EO7X47AA submitted 2025-07-22 stat.ML cond-mat.stat-mechcs.LGphysics.data-an

Roger Guimera , Marta Sales-Pardo This is my paper · ORCID
classification stat.MLcond-mat.stat-mechcs.LGphysics.data-an
keywords symbolicregressionBayesianmodelselectionminimumdescriptionlengthinformationcriterionaveraginglearnabilitytransitionLaplaceapproximationRashomonsets
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

This paper argues that symbolic regression—automatically discovering closed-form equations from data—should be built on the posterior probability of each candidate model rather than on heuristic loss, complexity, and exploration choices. It shows that under Gaussian noise and the Laplace approximation, the practical model-selection criterion is to minimize $L(m_i,D)=B_1(m_i,D)/2-\log p(m_i)$, a description length combining goodness of fit, a parameter penalty, and an explicit prior over expressions. The authors claim this criterion is consistent, Bayes-optimal, and free of ad hoc trade-offs, and they demonstrate on simple generated datasets that a standard heuristic implementation systematically overfits while the Bayesian criterion recovers the true model, or underfits only when noise is so high that the true model is undetectable. If the argument is right, symbolic regression should output a distribution over models rather than a single best equation, and prediction should be done by averaging over that distribution.

What carries the argument

The load-bearing object is the marginal likelihood $p(D|m_i)=\int_{\Theta_i} d\theta_i\, p(D|m_i,\theta_i)p(\theta_i|m_i)$, approximated by Laplace's method. Its negative logarithm plus the model prior defines the description length $L(m_i,D)$, which the paper interprets as the number of nats needed to convey the model and the data, and equivalently as the energy (information Hamiltonian) of a canonical ensemble of models at inverse temperature $\beta=1$. The score $B_1/2-\log p(m_i)$ replaces three heuristic choices of traditional symbolic regression: the loss, the complexity measure, and the Pareto-front elbow selection. The optional $B_2$ term uses the Fisher information matrix to reward models whose predictions are insensitive to small parameter changes, and the prior $p(m_i)$ is made explicit and transparent through a maximum-entropy distribution over operator counts.

What would settle it

Generate data from a model with a flat or multimodal likelihood surface, such as $y=\theta\sin(\omega x)+\epsilon$ with a broad prior over $\omega$ and small $N$, and compare the model ranking produced by $L(m_i,D)$ against the exact marginal likelihood computed by numerical quadrature; a substantial mismatch, or frequent selection of a wrong model where the exact posterior favors the true one, would falsify the claim that the Laplace-based criterion inherits the Bayes-optimality guarantees.

Watch

Extended reading notes

Core claim

The central claim is that the complete solution to symbolic regression is the posterior distribution $p(m_i|D)\propto \exp[-L(m_i,D)]$ over candidate closed-form models, and that this distribution can be computed under explicit approximations. Marginalizing each model's parameters with Laplace's method gives $L(m_i,D)=B_1(m_i,D)/2-\log p(m_i)$, where $B_1(m_i,D)=-2\log p(D|m_i,\hat\theta_i)+(k_i+1)\log N$ is the Bayesian information criterion; keeping one more term replaces $B_1$ with $B_2$, which adds $\log\det I(\hat\theta_i)$, the Fisher information at the maximum-likelihood parameters. The paper claims that this one score integrates goodness of fit, parametric complexity, and structural prior plausibility in a single set of units, so the most plausible model is $\arg\min_{m_i} L(m_i,D)$. It further claims that traditional heuristic symbolic regression is implicitly using a prior $\log p_{\mathrm{trad}}\approx \frac{k_i+1}{2}\log N$ that favors more complex models, which explains its dramatic overfitting even on a constant model, and that the Bayesian criterion is consistent as $N\to\infty$ because the marginal likelihood is extensive while the prior is intensive. Finally, it claims that accounting for uncertainty forces one to consider whole ensembles of models, with important consequences including model averaging and a fundamental learnable-to-unlearnable transition.

Load-bearing premise

The practical ranking by $L(m_i,D)$ assumes the Laplace approximation is accurate: for every candidate model, the likelihood must be sharply peaked around its best-fit parameters and the prior smooth there, and when that fails the computed score is not the true log posterior, so the stated consistency and optimality guarantees no longer apply to the implemented method.

Editorial extensions

If this is right

  • If the Bayesian criterion is adopted, any symbolic regression method minimizing a heuristic loss is implicitly using a prior that rewards complexity; minimizing $L(m_i,D)$ removes the need for ad hoc Pareto-front and elbow choices.
  • The posterior over models, not the single most plausible equation, is the complete output of symbolic regression; predictions should be made by Bayesian model averaging over posterior samples.
  • Because the marginal likelihood grows extensively with $N$ while the prior stays intensive, the criterion is consistent: as sample size grows it selects the true model with probability approaching one, as long as no true model has zero prior probability.
  • There is a finite-sample learnability transition: above a certain noise level the true generating model ceases to be the minimum of the description-length landscape, so no algorithm can recover it from data alone, and near that transition the set of near-optimal models (a Rashomon set) is diverse and mutually inconsistent.

Reading between the lines

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

  • Editorial inference: the same description-length objective could be grafted onto modern deep-learning symbolic regression by using $L(m_i,D)$ as the training or reward signal for a generator over expressions, turning a heuristic search into an approximately Bayesian sampler; the paper does not test this.
  • Editorial inference: because the Laplace approximation is the weakest step, comparing $B_1/B_2$ rankings against exact numerical marginal likelihoods on small expression spaces would quantify how often the implemented criterion diverges from the true posterior, and where the stated consistency guarantee actually holds.
  • Editorial inference: the learnability-transition result suggests a practical scientific diagnostic—reporting the spread of description lengths across the Rashomon set for a dataset would tell researchers whether the data are sufficient to pin down a unique law, rather than trusting one fitted equation.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper argues that symbolic regression should be founded on Bayesian probability rather than on heuristic objectives. It derives the model selection criterion L(m_i,D) = B1/2 - log p(m_i) (and the B2 variant with a Fisher-information term) from a Laplace approximation to the marginal likelihood, and then gives four arguments for the Bayesian approach: Cox's theorem, consistency, minimum description length, and Dutch-book coherence. It contrasts this with traditional symbolic regression (exemplified by PySR), reporting two synthetic experiments where the Bayesian Machine Scientist (BMS) outperforms PySR, and it discusses model averaging and Rashomon sets as consequences of working with the full posterior over models.

Significance. If the claims were fully established, the paper would provide a useful conceptual foundation for probabilistic symbolic regression and a clear argument that model ensembles, not single equations, are the appropriate output. The derivation of the BIC-type criterion from the Laplace approximation is standard and correctly presented, and the information-theoretic and statistical-physics interpretations are pedagogically valuable. The paper also makes a constructive point: traditional symbolic regression's complexity penalties can be seen as implicit and often unprincipled, whereas the Bayesian criterion makes all assumptions explicit. However, the paper overstates the guarantees it establishes: the strong consistency and optimality claims are proven only for the exact posterior under ideal conditions, not for the implemented BMS approximation, and the empirical evidence is a single run per condition with no repeated trials or code. These gaps are load-bearing because the paper's central argument is that the Bayesian approach 'provides guarantees of performance that heuristic approaches lack.'

major comments (4)
  1. [Section 4, consistency argument] The sentence 'any alternative that does not coincide with the Bayesian approach in this limit is virtually guaranteed to select the wrong model' is not justified by the preceding argument. Bayesian model selection consistency is known to hold only when the true model is contained in the model space with positive prior probability and under certain regularity conditions; the statement in Section 3a that the prior washes out 'as long as we do not assign p(m_i)=0' ignores the case where the true model is outside the search grammar or has zero prior under the chosen p(m). Without stating these conditions, the universal guarantee is unsubstantiated and should be weakened or qualified.
  2. [Section 2, Eqs. (2.5)-(2.9)] The consistency and Bayes-optimality arguments in Section 4 apply to the exact posterior p(m_i|D) in Eq. (2.4), but the implemented BMS uses the Laplace-approximated criteria in Eqs. (2.5)-(2.9). The assumptions that the likelihood is sufficiently peaked and the prior is smooth are stated, but the paper does not address model spaces where these fail: redundant parameters, near-collinear terms, and flat likelihood directions can make the Laplace integral dominated by boundaries or saddle points rather than a single interior peak, so the dropped O(1) terms are not negligible. Consequently, the advertised guarantees do not automatically transfer to the implemented BMS; the authors should either prove that the approximation is asymptotically valid for the model classes they consider or explicitly state that the guarantees hold for the exact posterior only.
  3. [Section 5, Figs. 1-2] The empirical demonstration consists of one dataset per (N,σ) cell, with no seeds, no repeated trials, and no code or configuration details beyond 'default parameters'. The statement that the example 'is sufficient to prove that the heuristics chosen do not lead to consistent model selection' is therefore too strong: a single run of PySR on a single dataset per condition cannot establish a general failure mode. The authors should provide repeated runs with error bars or at least rephrase these results as illustrative rather than proof.
  4. [Section 5, Eq. (5.4)] The assertion that 'the prior effectively being used by the traditional approach is log p_trad = (ki+1)/2 log N + ...' conflates a model-selection penalty with a Bayesian prior. A complexity penalty in a heuristic loss is not a prior over models, and the ellipsis hides the fact that the approximation terms in B1/B2 are part of the marginal likelihood, not a prior. The claim that traditional approaches 'favor a priori models with more, rather than fewer, parameters' is an analogy that holds only for this particular penalty form and should be presented as such, not as an identified prior.
minor comments (5)
  1. [Section 5, Eq. (5.1)] The coefficients α_o and β_o in the operator prior are not specified numerically; the paper cites previous work [14] but does not state the values or how they were estimated, which makes the prior non-reproducible from this manuscript alone.
  2. [Section 1, desideratum (ii)] The paper promises 'no ad hoc parameters or thresholds need to be fixed' to balance fit and complexity, but the prior in Eq. (5.1) contains parameters α_o and β_o fitted to a corpus. While these are not balancing parameters in the same sense, the text should acknowledge that the prior still requires a choice of corpus and estimation procedure.
  3. [Section 6(a)] Line 'consider a situation in which one whats to predict' contains a typo: 'whats' should be 'wants'.
  4. [Section 4, Cox argument] The statement that any reasoning system not conforming to Eq. (2.4) 'must violate some of the very basic commonsensical conditions assumed by Cox' is too categorical, given known limitations of Cox's theorem (e.g., requiring real-valued plausibilities and infinite precision). A softer formulation under the standard Cox axioms would be more accurate.
  5. [Section 6(b)] The learnability transition and the claim that no algorithm can identify the true model above a certain noise level are cited to [16] but not derived or illustrated in this manuscript; for a self-contained perspective, a brief informal derivation or a reference to the precise statement in [16] would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation of the Bayesian description-length criterion is self-contained, and self-citations are illustrative or point to prior results with independent derivations.

full rationale

The central derivation is not circular. Section 2 starts from Bayes' theorem (Eq. 2.2) and the Gaussian likelihood (Eq. 2.3), marginalizes over parameters (Eq. 2.4), and then obtains L(mi,D)=B1/2-log p(mi) via the Laplace approximation under explicitly stated conditions (Eqs. 2.5-2.7), with the optional B2 correction (Eqs. 2.8-2.9). This is a standard Bayesian model-selection derivation, and the objective is not defined in terms of the outputs it later explains. The consistency and Bayes-optimality arguments in Section 4 rely on external axiomatizations (Cox, de Finetti) and on the standard extensivity/intensity argument for the marginal likelihood versus the prior; the paper even acknowledges the needed caveat that no model may be assigned zero prior. The empirical comparisons in Section 5 use the authors' own BMS implementation (Ref. 14), but the BMS outputs are not fitted to the two synthetic datasets shown, and the prior in Eq. (5.1) is explicitly labeled as an arbitrary but transparent choice. Claims about the learnability transition and Rashomon sets in Section 6 cite prior published work (Refs. 14, 15, 16) that contains the derivations; citing one's own earlier proofs is not circular unless the present argument reduces to the citation, which it does not. The main practical risk, that the Laplace approximation may be inaccurate for some candidate models so that the implemented B1/B2 criterion is not exactly the log posterior, is an approximation-validity limitation rather than a circularity, and the paper presents that approximation as an explicit assumption.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The argument rests on standard Bayesian machinery: a generative model (Eq. 2.1), the Laplace approximation (Section 2), and axioms of probability (Cox, de Finetti, Section 4). The only fitted parameters in the implemented prior are α_o and β_o from the authors' earlier BMS work. No new particles, fields, forces, or dimensions are introduced.

free parameters (1)
  • α_o, β_o (operator prior coefficients) = Not given in this paper; fitted to a corpus of formulas in ref [14]
    The prior in Eq. (5.1), p(mi) = exp[-Σ_o (α_o n_oi + β_o n_oi^2)], is used by the BMS implementation in the experiments. Its coefficients were fitted to an empirical corpus in earlier work by the same group, so they are data-dependent inputs, not constants forced by the problem.
assumptions (4)
  • domain assumption The observed data are generated by a true closed-form model m* in the candidate space M plus independent Gaussian noise with unknown variance (Eq. 2.1).
    This generative assumption underlies the likelihood in Eq. (2.3) and all subsequent posterior and consistency arguments; if the true model is not in M or noise is not Gaussian, the guarantees do not transfer.
  • domain assumption The Laplace approximation is accurate: likelihood sufficiently peaked around the MLE and prior smooth, for every model in M (Section 2).
    The computable criterion L(mi,D) in Eqs. (2.5) and (2.8) is only the exact log posterior under this approximation; the Bayes-optimality and consistency guarantees are stated for the exact posterior.
  • standard math Cox's theorem and de Finetti's Dutch book theorem justify probability as the unique consistent framework for degrees of belief (Section 4).
    These are standard results from probability theory, cited to Refs. [22-24], used to argue that any non-Bayesian model plausibility assignment violates basic consistency.
  • ad hoc to paper The maximum-entropy operator prior of Eq. (5.1) is a reasonable representation of prior plausibility over scientific formulas.
    The coefficients are fitted to an empirical corpus in prior work [14]; the paper acknowledges the choice is arbitrary but explicit, and the BMS experimental results depend on it.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Bayesian symbolic regression: Automated equation discovery from a physicists' perspective." pith.science (2026). https://pith.science/paper/EO7X47AA

@misc{pith2026250719540,
  author       = {Pith},
  title        = {Pith review of: Bayesian symbolic regression: Automated equation discovery from a physicists' perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EO7X47AA}},
  note         = {Machine review of arXiv:2507.19540}
}
read the original abstract

Symbolic regression automates the process of learning closed-form mathematical models from data. Standard approaches to symbolic regression, as well as newer deep learning approaches, rely on heuristic model selection criteria, heuristic regularization, and heuristic exploration of model space. Here, we discuss the probabilistic approach to symbolic regression, an alternative to such heuristic approaches with direct connections to information theory and statistical physics. We show how the probabilistic approach establishes model plausibility from basic considerations and explicit approximations, and how it provides guarantees of performance that heuristic approaches lack. We also discuss how the probabilistic approach compels us to consider model ensembles, as opposed to single models.

Figures

Figures reproduced from arXiv: 2507.19540 by the authors.

Figure 1
Figure 1. For varying number of data points N ∈ {10, 100, 1000} and different levels of observational noise σ ∈ {0.05, 0.5, 5, 50}, we generate data (gray symbols) through the process y k = θ ∗ 0 + θ ∗ 1 x + ϵ k, with θ ∗ 0 = −2.3 and θ ∗ 1 = 4.1. We then use traditional heuristic symbolic regression (using PySR [7] with default parameters; blue lines) and Bayesian symbolic regression (using the Bayesian machine scientist, BM… view at source ↗
Figure 2
Figure 2. For varying number of data points N ∈ {10, 100, 1000} and different levels of observational noise σ ∈ {0.05, 0.5, 5, 50}, we generate data (gray symbols) through the process y k = θ ∗ 0 + ϵ k, with θ ∗ 0 = 31. We then use traditional heuristic symbolic regression (using PySR [7] with default parameters; blue lines) and Bayesian symbolic regression (using the Bayesian machine scientist, BMS [14]; orange lines) to lea… view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 24 canonical work pages

  1. [16]

    2023 Fundamental limits to learning closed-form mathematical models from data

    Fajardo-Fontiveros O, Reichardt I, De Los Ríos HR, Duch J, Sales-Pardo M, Guimerà R. 2023 Fundamental limits to learning closed-form mathematical models from data. Nat. Comm. 14, 1043

    work page 2023

  2. [1]

    2023 Scientific discovery in the age of artificial intelligence

    Wang H, Fu T, Du Y, Gao W, Huang K, Liu Z, Chandak P , Liu S, Van Katwyk P , Deac A, Anandkumar A, Bergen K, Gomes CP , Ho S, Kohli P , Lasenby J, Leskovec J, Liu TY, Manrai A, Marks D, Ramsundar B, Song L, Sun J, Tang J, Veliˇ ckovi´ c P , Welling M, Zhang L, Coley CW, Bengio Y, Zitnik M. 2023 Scientific discovery in the age of artificial intelligence. N...

    doi:10.1038/s41586-023-06221-2 2023

  3. [2]

    2023 Combining data and theory for derivable scientific discovery with AI-Descartes

    Cornelio C, Dash S, Austel V , Josephson TR, Goncalves J, Clarkson KL, Megiddo N, El Khadir B, Horesh L. 2023 Combining data and theory for derivable scientific discovery with AI-Descartes. Nat. Commun. 14, 1777. 4After the movie Rashomon, in which Akira Kurosawa concatenates a series of contradictory accounts of the same crime. 11royalsocietypublishing.o...

    work page 2023

  4. [3]

    1987 Scientific Discovery: Computational Explorations of the Creative Processes

    Langley PW, Simon HA, Bradshaw G, Zytkow JM. 1987 Scientific Discovery: Computational Explorations of the Creative Processes. The MIT Press, Cambridge

    work page 1987

  5. [4]

    2007 Computational Discovery of Scientific Knowledge

    Džeroski S, Todorovski L, editors. 2007 Computational Discovery of Scientific Knowledge. Lecture Notes in Artificial Intelligence. Springer

    work page 2007

  6. [5]

    1992 Genetic programming: On the programming of computers by means of natural selection

    Koza JR. 1992 Genetic programming: On the programming of computers by means of natural selection. Cambridge, MA: MIT Press

    work page 1992

  7. [6]

    2009 Distilling free-form natural laws from experimental data

    Schmidt M, Lipson H. 2009 Distilling free-form natural laws from experimental data. Science 324, 81–5. (10.1126/science.1165893)

    doi:10.1126/science.1165893 2009

  8. [7]

    2023 Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl

    Cranmer M. 2023 Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl. arXiv:2305.01582 (10.48550/arXiv.2305.01582)

    arXiv doi:10.48550/arxiv.2305.01582 2023

Show all 30 references
  1. [8]

    2016 Discovering governing equations from data by sparse identification of nonlinear dynamical systems

    Brunton SL, Proctor JL, Kutz JN. 2016 Discovering governing equations from data by sparse identification of nonlinear dynamical systems. Proc. Natl. Acad. Sci. U.S.A. 113, 3932–3937

    2016

  2. [9]

    2021 Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients

    Petersen BK, Larma ML, Mundhenk TN, Santiago CP , Kim SK, Kim JT. 2021 Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients. In International Conference on Learning Representations

    2021

  3. [10]

    2023 Efficient generator of mathematical expressions for symbolic regression

    Mežnar S, Džeroski S, Todorovski L. 2023 Efficient generator of mathematical expressions for symbolic regression. Mach. Learn. 112, 4563–4596

    2023

  4. [11]

    2020 AI Feynman: A physics-inspired method for symbolic regression

    Udrescu SM, Tegmark M. 2020 AI Feynman: A physics-inspired method for symbolic regression. Sci. Adv. 6, eaay2631

    2020

  5. [12]

    2021 Contemporary symbolic regression methods and their relative performance

    La Cava W, Burlacu B, Virgolin M, Kommenda M, Orzechowski P , de França FO, Jin Y, Moore JH. 2021 Contemporary symbolic regression methods and their relative performance. Adv. Neural Inf. Process. Syst. 2021, 1

    2021

  6. [13]

    2024 Interpretable scientific discovery with symbolic regression: a review

    Makke N, Chawla S. 2024 Interpretable scientific discovery with symbolic regression: a review. Artificial Intelligence Review 57, 2. (10.1007/s10462-023-10622-0)

    2024 doi

  7. [14]

    2020 A Bayesian machine scientist to aid in the solution of challenging scientific problems

    Guimerà R, Reichardt I, Aguilar-Mogas A, Massucci FA, Miranda M, Pallarès J, Sales-Pardo M. 2020 A Bayesian machine scientist to aid in the solution of challenging scientific problems. Sci. Adv. 6, eaav6971

    2020

  8. [15]

    2020 Bayesian machine scientist to compare data collapses for the Nikuradse dataset

    Reichardt I, Pallarès J, Sales-Pardo M, Guimerà R. 2020 Bayesian machine scientist to compare data collapses for the Nikuradse dataset. Phys. Rev. Lett. 124, 084503

    2020

  9. [17]

    1978 Estimating the dimension of a model

    Schwarz G. 1978 Estimating the dimension of a model. Ann. Stat. 6, 461–464

    1978

  10. [18]

    2010 Bayesian model selection and statistical modeling

    Ando T. 2010 Bayesian model selection and statistical modeling. CRC Press

    2010

  11. [19]

    2024 Exhaustive Symbolic Regression

    Bartlett DJ, Desmond H, Ferreira PG. 2024 Exhaustive Symbolic Regression. IEEE Transactions on Evolutionary Computation 28, 950–964. (10.1109/TEVC.2023.3280250)

    2024

  12. [20]

    2007 The Minimum Description Length Principle

    Grünwald PD. 2007 The Minimum Description Length Principle. Cambridge, Massachusetts: The MIT Press

    2007

  13. [21]

    2019 Information theory for fields

    Enßlin TA. 2019 Information theory for fields. Annalen der Physik 531, 1800127. (10.1002/andp.201800127)

    2019 doi

  14. [22]

    1946 Probability, frequency and reasonable expectation

    Cox RT. 1946 Probability, frequency and reasonable expectation. Am. J. Phys. 14, 1–10

    1946

  15. [23]

    2003 Probability Theory: The Logic of Science

    Jaynes ET. 2003 Probability Theory: The Logic of Science. Cambridge University Press

    2003

  16. [24]

    2022 Dutch Book Arguments

    Vineberg S. 2022 Dutch Book Arguments. In Zalta EN, Nodelman U, editors, The Stanford Encyclopedia of Philosophy , . Metaphysics Research Lab, Stanford University Fall 2022 edition

    2022

  17. [25]

    2023 Priors for symbolic regression

    Bartlett D, Desmond H, Ferreira P . 2023 Priors for symbolic regression. In Proceedings of the Companion Conference on Genetic and Evolutionary Computation GECCO ’23 Companion p. 2402–2411 New York, NY, USA. Association for Computing Machinery. (10.1145/3583133.3596327)

    2023

  18. [26]

    2024 Incorporating background knowledge in symbolic regression using a computer algebra system.Machine Learning: Science and Technology 5, 025057

    Fox C, Tran ND, Nacion FN, Sharlin S, Josephson TR. 2024 Incorporating background knowledge in symbolic regression using a computer algebra system.Machine Learning: Science and Technology 5, 025057. (10.1088/2632-2153/ad4a1e)

    2024 doi

  19. [27]

    1999 Bayesian model averaging: A tutorial

    Hoeting JA, Madigan D, Raftery AE, Volinsky CT. 1999 Bayesian model averaging: A tutorial. Stat. Sci. 14, 382–417

    1999

  20. [28]

    2019 Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead

    Rudin C. 2019 Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nat. Mach. Intell. 1, 206–215

    2019

  21. [29]

    2025 Human mobility is well described by closed-form gravity-like models learned automatically from data.Nat

    Cabanas-Tirapu O, Danús L, Moro E, Sales-Pardo M, Guimerà R. 2025 Human mobility is well described by closed-form gravity-like models learned automatically from data.Nat. Comm. 16, 1336. 12royalsocietypublishing.org/journal/rsta Phil. T rans. R. Soc. A 0000000

    2025

  22. [30]

    2024 Artificial intelligence needs a scientific method-driven reset

    Amaral LAN. 2024 Artificial intelligence needs a scientific method-driven reset. Nat. Phys. 20, 523–524

    2024

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.