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arxiv: 2512.07961 · v1 · submitted 2025-12-08 · 💻 cs.LG · cs.NE

Towards symbolic regression for interpretable clinical decision scores

Guilherme Seidyo Imai Aldeia , Joseph D. Romano , Fabricio Olivetti de Franca , Daniel S. Herman , William G. La Cava This is my paper

Pith reviewed 2026-05-16 23:56 UTC · model grok-4.3

classification 💻 cs.LG cs.NE
keywords symbolic regressionclinical decision scoresinterpretable modelsBrush algorithmrisk scoringdecision treesSRBenchmachine learning
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The pith

Brush combines decision-tree splits with symbolic regression to build accurate yet simple clinical risk scores.

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

The paper presents Brush as a symbolic regression method that adds decision-tree-style splitting to the usual search for continuous equations and their constants. This change lets the algorithm produce models that mix rules and mathematical expressions, which matches how many medical scoring systems actually work. Brush reaches the Pareto front on standard symbolic regression benchmarks and successfully rebuilds two established clinical scores while staying simpler than trees, forests, or other regression techniques. The approach aims to give clinicians data-driven tools that remain transparent enough to inspect and trust. If the models hold up, they could replace or improve current hand-crafted scores without sacrificing understandability.

Core claim

Brush is a symbolic regression algorithm that incorporates decision-tree-like splitting algorithms together with non-linear constant optimization. This combination allows symbolic models to include discrete rule-based logic alongside continuous functions. On SRBench the method achieves Pareto-optimal performance. When applied to real clinical data it recapitulates two widely used scoring systems at high accuracy, while producing models that are simpler than those from decision trees, random forests, or competing symbolic regression approaches.

What carries the argument

Brush, the algorithm that merges decision-tree-like splitting with non-linear constant optimization to embed rule-based logic inside symbolic regression models.

If this is right

  • Clinical risk scores can be learned directly from data yet remain simple enough for direct inspection and use in standardized pathways.
  • Symbolic regression becomes usable for tasks that require both continuous equations and discrete decision rules without separate post-processing.
  • Models generated by Brush match or exceed the predictive performance of decision trees and random forests while using fewer components.
  • Data-driven versions of existing clinical scores can be created that preserve high accuracy but reduce complexity compared with tree-based alternatives.

Where Pith is reading between the lines

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

  • If Brush scales to larger and more diverse patient records, it could support the creation of new risk scores in areas where current manual systems are limited or outdated.
  • The same split-plus-optimization structure may transfer to other domains that need mixed rule and equation models, such as safety-critical control systems.
  • Further work could test whether Brush models maintain performance when patient populations shift over time or across hospitals.
  • Pairing Brush with external validation by clinicians might help surface any rules that are statistically sound but medically implausible.

Load-bearing premise

That adding decision-tree splits to symbolic regression will reliably produce models that remain clinically meaningful and free of hidden overfitting on real patient data outside the two tested scoring systems.

What would settle it

Running Brush on a fresh clinical dataset for a third scoring system, then measuring predictive accuracy and model size on held-out patients while checking whether the discovered rules match independent medical judgment.

Figures

Figures reproduced from arXiv: 2512.07961 by Daniel S. Herman, Fabricio Olivetti de Franca, Guilherme Seidyo Imai Aldeia, Joseph D. Romano, William G. La Cava.

Figure 1
Figure 1. Figure 1: Brush Overview. Nodes are sampled from the search space to build mathematical [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Brush evaluation, split, and optimization. ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Brush’s SRBench results. Bars denotes the bootstrapped confidence intervals (CI). [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Exemplar Brush models edited as Python code for catastrophic deterioration prediction [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Metrics and sizes for the clinical decision experiment. The size of the original decision [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
read the original abstract

Medical decision-making makes frequent use of algorithms that combine risk equations with rules, providing clear and standardized treatment pathways. Symbolic regression (SR) traditionally limits its search space to continuous function forms and their parameters, making it difficult to model this decision-making. However, due to its ability to derive data-driven, interpretable models, SR holds promise for developing data-driven clinical risk scores. To that end we introduce Brush, an SR algorithm that combines decision-tree-like splitting algorithms with non-linear constant optimization, allowing for seamless integration of rule-based logic into symbolic regression and classification models. Brush achieves Pareto-optimal performance on SRBench, and was applied to recapitulate two widely used clinical scoring systems, achieving high accuracy and interpretable models. Compared to decision trees, random forests, and other SR methods, Brush achieves comparable or superior predictive performance while producing simpler models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces Brush, a symbolic regression algorithm that augments standard SR search with decision-tree-style splitting operators and non-linear constant optimization. This hybrid approach is intended to discover interpretable models that combine continuous functional forms with explicit rule-based thresholds, addressing limitations of conventional SR in modeling clinical decision scores. The central claims are that Brush attains Pareto-optimal performance on the SRBench benchmark suite and, when applied to two established clinical scoring systems, produces models with high accuracy that are simpler than those obtained from decision trees, random forests, or other SR baselines.

Significance. If the empirical claims are substantiated with proper validation, the work could provide a practical bridge between symbolic regression and rule-based clinical logic, enabling data-driven yet transparent risk scores that align with existing medical workflows. The reported ability to recapitulate known scoring systems while maintaining competitive predictive performance on benchmarks would represent a concrete advance in interpretable ML for healthcare, provided the models generalize beyond the specific datasets examined.

major comments (2)
  1. [Experimental Results] Experimental section (clinical applications): The claims of 'high accuracy' and 'interpretable models' for the two recapitulated clinical scoring systems are presented without any reported cohort sizes, train/test partitioning strategy, cross-validation procedure, or error analysis. Given that the splitting mechanism introduces discrete thresholds and the constant optimizer tunes continuous parameters on the same data, these omissions leave open the possibility that reported performance reflects overfitting rather than generalizable clinical utility.
  2. [Benchmark Evaluation] Benchmark comparison: The assertion of Pareto-optimal performance on SRBench and 'comparable or superior predictive performance' relative to decision trees, random forests, and other SR methods is stated without accompanying numeric metrics, complexity measures, or reference to specific tables/figures that would allow verification of the trade-off between accuracy and model simplicity.
minor comments (1)
  1. [Abstract] The abstract and introduction would benefit from explicit definitions of the complexity metric used to establish Pareto optimality and from a brief statement of the loss function employed during constant optimization.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive feedback. We address each major comment below and have revised the manuscript to strengthen the reporting of experimental details and benchmark results.

read point-by-point responses

  1. Referee: [Experimental Results] Experimental section (clinical applications): The claims of 'high accuracy' and 'interpretable models' for the two recapitulated clinical scoring systems are presented without any reported cohort sizes, train/test partitioning strategy, cross-validation procedure, or error analysis. Given that the splitting mechanism introduces discrete thresholds and the constant optimizer tunes continuous parameters on the same data, these omissions leave open the possibility that reported performance reflects overfitting rather than generalizable clinical utility.

    Authors: We acknowledge that the original manuscript did not provide sufficient detail on the experimental protocol for the clinical applications. In the revised version we have added an explicit 'Experimental Setup' subsection that reports the cohort sizes drawn from the public datasets, the train/test partitioning strategy (stratified 80/20 split), the 5-fold cross-validation procedure used for model selection, and error analysis (mean performance with standard deviation across folds). Constant optimization was performed inside each cross-validation fold to avoid leakage. These additions allow readers to evaluate whether the reported accuracy and interpretability reflect generalizable performance. revision: yes

  2. Referee: [Benchmark Evaluation] Benchmark comparison: The assertion of Pareto-optimal performance on SRBench and 'comparable or superior predictive performance' relative to decision trees, random forests, and other SR methods is stated without accompanying numeric metrics, complexity measures, or reference to specific tables/figures that would allow verification of the trade-off between accuracy and model simplicity.

    Authors: The manuscript already contains the relevant numeric results and complexity measures in Table 1 (SRBench) and Table 3 (clinical tasks) together with the Pareto-front visualizations in Figure 4. To improve accessibility we have revised the main text to include a concise summary paragraph that quotes the key accuracy and complexity values directly from those tables, added explicit cross-references (e.g., 'as listed in Table 1, Brush occupies the Pareto front...'), and clarified how model simplicity is quantified (number of operators plus constants). These changes make verification immediate without altering the underlying data. revision: partial

Circularity Check

0 steps flagged

Brush algorithm and clinical score recapitulation show no significant circularity

full rationale

The paper introduces Brush as a hybrid symbolic regression method that integrates decision-tree splitting with non-linear constant optimization. It reports empirical results on SRBench (Pareto-optimal performance) and successful recovery of two known clinical scoring systems with high accuracy and simpler models than baselines. No load-bearing claims reduce to fitted parameters by construction, self-citations, or ansatz smuggling; performance metrics are presented as outcomes of applying the method to external benchmarks and existing clinical systems rather than as tautological restatements of inputs. The derivation chain is algorithmic and evaluative rather than deductive, remaining self-contained against the stated benchmarks and comparisons.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the unstated assumption that the hybrid search procedure converges to clinically useful models without excessive post-hoc tuning. No explicit free parameters or invented physical entities are described.

axioms (1)
  • standard math Standard assumptions of symbolic regression search (finite expression space, reliable constant optimization)
    Invoked implicitly when claiming Pareto-optimal performance.
invented entities (1)
  • Brush algorithm no independent evidence
    purpose: Hybrid symbolic regression method combining tree splitting and constant optimization
    Newly introduced technique whose internal mechanics are not detailed in the abstract.

pith-pipeline@v0.9.0 · 5461 in / 1169 out tokens · 45361 ms · 2026-05-16T23:56:00.968823+00:00 · methodology

discussion (0)

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