Uncertainty Quantification in the Tsetlin Machine

Alex Yakovlev; Kunal Ganesh Dumbre; Lei Jiao; Mayur Kishor Shende; Ole-Christoffer Granmo; Rishad Shafik; Runar Helin; Vladimir I. Zadorozhny

REVIEW 5 major objections 5 minor 20 references

Uncertainty Quantification in the Tsetlin Machine

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

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Pith's one-line read A linear formula turns any Tsetlin machine's class sum into a probability score.

desk verdict A useful heuristic for TM uncertainty, but the central calibration claim collapses when negative-polarity clauses are included. read the letter →

arxiv 2507.04175 v1 pith:5CYT4WXD submitted 2025-07-05 cs.LG cs.AI

Runar Helin , Ole-Christoffer Granmo , Mayur Kishor Shende , Lei Jiao , Vladimir I. Zadorozhny , Kunal Ganesh Dumbre , Rishad Shafik , Alex Yakovlev This is my paper · ORCID

classification cs.LGcs.AI

keywords TsetlinmachineuncertaintyquantificationprobabilityscoreclasssumselectivepredictionexplainabilityCIFAR-10feedbackequilibrium

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

Tsetlin machines classify by counting logical clauses that vote for and against a target, and the difference of those counts is the class sum $v(x)$, clipped to $[-T,T]$. This paper claims that $v(x)$ carries calibrated probability information: at learning equilibrium, the probability that a clause receives reinforcing feedback equals the probability it receives correcting feedback, and solving that balance gives $P(y=1|x)=\frac{1}{2}(1+v(x)/T)$. If correct, any Tsetlin machine already outputs a probability score at inference time without retraining or extra calibration. The paper tests this on simulated data and on CIFAR-10 image classification, where thresholding the normalized score at 0.6 yields 93.2% accuracy on 1,231 high-certainty test samples. The practical payoff is a built-in certainty signal for a transparent rule-based model, enabling selective prediction and more human-readable explanations.

What carries the argument

The central object is the probability score $P(y=1|x)=\frac{1}{2}(1+v(x)/T)$, where $v(x)$ is the weighted class sum (matching positive clauses minus matching negative clauses) and $T$ is the target value that clips the class sum. The derivation mechanism is the equilibrium condition $P(I|x)=P(II|x)$: setting the probabilities of a clause receiving Type I versus Type II feedback equal, then solving for $v(x)$, yields $v=T[2P(y=1|x)-1]$, which inverts to the probability score. Since the class sum is already clipped to $[-T,T]$, the same map is also the most natural way to rescale the class sum to a probability. This machinery carries the argument by turning a learning-dynamics balance into an inference-time probability estimate.

What would settle it

Run the Section 3.1 single-pattern noise experiment and record, per epoch, the empirical frequencies with which the pattern receives Type I and Type II feedback; if those frequencies do not approach equality even once the class sum has reached its stationary oscillation, the balance condition behind Eq. (7) is violated and the probability score's theoretical grounding fails.

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Extended reading notes

Core claim

The central discovery is that the class sum is not just a voting tally but a probability surrogate. The paper derives $P(y=1|x)=\frac{1}{2}(1+v(x)/T)$ by setting the total probability of Type I feedback for a positive clause equal to the total probability of Type II feedback for that clause, then solving for $v(x)$. The same formula is the natural affine rescaling of the clipped class sum to the unit interval, so it also holds as a scaling identity even without the equilibrium argument. The authors claim the score is inherent to any Tsetlin machine variant, applies to multiclass problems after normalizing by the sum of per-class scores, and behaves sensibly outside the training domain, producing near-0.5 scores instead of overconfident extrapolation.

Load-bearing premise

The argument assumes that during training the chance of a clause receiving reinforcing feedback and the chance of it receiving correcting feedback become equal at equilibrium; the paper asserts this balance rather than proving it from the automaton update rules, and the plotted class sums keep oscillating instead of settling at a fixed value.

Editorial extensions

If this is right

Any trained Tsetlin machine can emit a probability score for each prediction from its class sum and $T$ alone; no calibration set or temperature scaling is needed.
The score provides an abstention rule: with a chosen threshold, only predictions above it are accepted. On the CIFAR-10 model, thresholding at normalized score 0.6 raises accuracy from 65% to 93.2% on the 1,231 accepted samples.
Outside the training-data region, binary-class scores fall back toward 0.5, so the model signals low confidence in extrapolation rather than overconfident predictions as neural networks often do.
For multiclass problems, the raw per-class score overstates certainty because it reflects one-vs-rest confidence; the paper's normalized score, the largest class score divided by the sum of all class scores, is the more honest certainty measure.
Because the score needs no true labels, it can be averaged over several epochs during training to smooth the oscillation of class sums, at the cost of slower inference.

Reading between the lines

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

A natural next test, not performed in the paper, is to benchmark the probability score as an out-of-distribution detector: the near-0.5 score outside the training domain suggests that low-scoring or near-0.5 samples can flag inputs far from the training data.
The alternative derivation as a pure affine rescaling means Eq. (7) can be viewed as an implicit calibration layer; one could combine it with distribution-free post-processing to produce coverage guarantees for selective prediction, which the paper does not explore.
Because the score is a deterministic function of the class sum, it should transfer to any Tsetlin machine variant that maintains a class sum, but the paper asserts this without a separate calibration experiment for each variant; a cross-variant study would settle the claim.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

You should know two things about 2507.04175. First, it proposes a clean, simple way to turn Tsetlin machine class sums into a [0,1] score: P = 0.5(1 + v/T). That explicit formula is new for TMs. Second, the paper's claim that this is a calibrated probability "derived through an analysis of the TM learning dynamics" does not hold up. The derivation in Sec. 2.2 balances Type I and Type II feedback only for positive-polarity clauses. For negative-polarity clauses, the balance condition is v = -T(2P(y=1|x)-1), the opposite sign. Since v is a single global class sum, both equilibria cannot exist except in the degenerate case. A drift calculation confirms this: with symmetric clause counts, the expected change in v is proportional to 2p-1, independent of v, so Eq. (6) is not a fixed point. The paper's own Fig. 1 and Sec. 4 show class sums oscillating, which is consistent with the missing balance.

What is genuinely useful: the linear score is a natural normalization of the clipped class sum, and the paper says so explicitly in Sec. 2.3. The simulations show a monotone relationship between average class sum and underlying noise, which is intuitive; the CIFAR-10 threshold experiment shows that filtering by the normalized score boosts accuracy from 65% to 93% on the most confident 12% of samples. That is a reasonable heuristic for selective prediction.

The soft spots are the central derivation, plus the empirical gaps: the key quantitative claim (that scores match noise levels) is literally "not shown here"; there are no error bars, no baselines, no calibration metrics like ECE or Brier, and no code. The CIFAR-10 normalized score (max score divided by sum) is introduced ad hoc and is not derived. None of these are fatal for a heuristic, but they are fatal for the abstract's assertion that the score is an inherent calibrated property.

My take: the paper is a modest extension with a useful heuristic, but it overclaims. The equilibrium derivation should be dropped or corrected, and the empirical section needs error bars, calibration curves, and released code.

Recommendation: send to peer review. The idea is worth a serious round of review, and the authors have been honest about limitations; with a major revision this could become a decent contribution to the TM literature.

Referee Report

5 major / 5 minor

Summary. The paper proposes a probability score for Tsetlin Machine (TM) predictions, P(y=1|x) = 1/2 (1 + v(x)/T), derived by balancing Type I and Type II feedback probabilities in Section 2.2. It claims this score is an inherent property of any TM variant. Experiments on simulated data show class sums roughly track noise levels, a two-feature moons visualization shows confidence decreasing outside the training domain, and a CIFAR-10 study uses a normalized multiclass score to show that thresholding on the score can improve accuracy on selected subsets. The paper concludes with limitations, including the acknowledged oscillation of class sums.

Significance. If the central derivation were sound, the paper would contribute a parameter-free calibration map from TM class sums to probabilities, with practical value for selective prediction and interpretability. The paper also ships a falsifiable empirical pattern -- probabilities outside the training domain approach 0.5 -- which is interesting if reproducible. However, the significance is conditional on the central claim, which is not established in the manuscript. The paper is honest about the oscillation limitation and the heuristic nature of the multiclass normalization, which narrows what is being claimed.

major comments (5)

[Section 2.2, Eqs. (4)-(6)] The derivation of Eq. (6) balances P(I|x) and P(II|x) for a positive-polarity clause using P(I|x) = P_I(x) P(y=1|x) and P(II|x) = P_II(x) P(y=0|x). The text notes that for a negative-polarity clause the two probabilities are opposite, but it never uses this fact. For a negative clause, the balance condition is P(I|x) = P_I(x) P(y=0|x) and P(II|x) = P_II(x) P(y=1|x), which yields the opposite equilibrium v = -T[2P(y=1|x)-1]. Since v(x) is a single class sum, both conditions can hold only in the degenerate case v=0 and P(y=1|x)=1/2. Thus Eq. (7) cannot be the equilibrium of a two-polarity TM under the paper's own feedback model, and the claim that it is 'derived through an analysis of the TM learning dynamics' is not supported.
[Section 4, Limitations; Figure 1] The paper itself acknowledges that the class sum 'oscillates and changes from epoch to epoch' rather than converging to a fixed point. That acknowledged behavior is in tension with the equilibrium derivation of Eq. (6), which assumes a stationary balance of feedback probabilities. The paper does not show that the time-averaged class sum is the fixed point of the clause-update dynamics, so the connection between the oscillation average and Eq. (7) is asserted rather than demonstrated.
[Section 2.3, Eq. (7)] The alternative derivation in Section 2.3 is purely a rescaling and shifting of the clipped class sum to [0,1]. This confirms that Eq. (7) is the natural normalization of v(x), but it does not establish that the resulting value is a calibrated probability. The claim that the probability score is 'coincidentally the same probability as receiving Type II feedback' is also not meaningful for calibration unless the feedback balance is proven. Without independent evidence, Eq. (7) remains a heuristic score, not a derived probability.
[Section 3.1, Figure 1; Section 3.2.2, Figures 5-7] The main empirical support for the probability-score interpretation is the statement that 'when using Eq. (7) on the average of the last 200 epochs, the resulting probability scores are approximately the same as the noise levels (not shown here).' This is a central quantitative claim, but no plot, table, or numeric error is given, and no error bars are provided for the single-pattern or CIFAR experiments. Similarly, Figure 7 reports accuracies for thresholded subsets without confidence intervals; with only 10,000 test samples and 65% base accuracy, the 0.932 figure needs a standard error or bootstrap interval to be assessable.
[Section 3.2.2, normalized score definition] The proposed multiclass normalized score (max individual probability divided by the sum of the ten individual probability scores) is introduced without a derivation or justification beyond a qualitative rationale. Its validity as a certainty measure is not connected to the learning dynamics, and no comparison with alternatives (e.g., softmax of the class sums, margin score, or entropy) is given. Since the CIFAR-10 conclusions rely entirely on this score, the paper's main application result is not fully supported.

minor comments (5)

[Section 3.1.1, Figure 3] The color scale for Figure 3 is not described in the text, and the overlap region and the outside-domain regions are identified by eye; a contour line at 0.5 or a quantitative comparison of the TM and MLP extrapolation behavior would make the claimed contrast precise.
[Section 3.2.2, CIFAR-10 setup] The hyperparameters for the two s-values (s=20 and s=1.2) are given in the text, but it is not stated how many runs were performed or whether the reported 65% and 63% accuracies are single runs; state the number of seeds or runs.
[Section 2.1, Eq. (3)] The weighted class sum is introduced but the weight update rule is never specified. Since the subsequent theory uses v(x) in the feedback formulas as if the weights were subsumed, a sentence clarifying the relationship between weights and the clipping to [-T,T] would be helpful.
[Throughout] Typographical and grammatical issues include 'tends to decrease the class sum' (singular/plural agreement), 'the the accuracy', 'with with different degree', and the incomplete sentence in the abstract's surrounding text; a careful proofread is needed.
[References] The reference list misses recent uncertainty-aware TM work that could provide a comparison baseline, and the paper would benefit from citing the formal convergence results of Jiao et al. and Zhang et al. in the derivation section, since those results concern feedback dynamics that are directly relevant to the equilibrium assumption.

Circularity Check

2 steps flagged · score 6.0 of 10

The probability score of Eq. (7) is the affine normalization of the clipped class sum; the 'learning-dynamics derivation' re-inserts the same affine function through the feedback probability PII and ignores negative-polarity clauses.

self definitional [Section 2.1-2.2, Eq. (7) and the feedback probability PII]
"If the sample does not belong to the target class, the clause is given reject feedback with probability PII(xi)=1/2(1+v(xi)/T). ... From this equation, one find that the probability P(y=1|x) for the target to be true given a sample can be estimated by the equation: P(y=1|x)=1/2(1+v(xi)/T), which is coincidentally the same probability as receiving Type II feedback."

PII is introduced as 1/2(1+v/T), and Eq. (7) is literally the same function of v. Since PI=1-PII, the equilibrium condition P(I|x)=P(II|x) reduces algebraically to (1-PII)P(y=1|x)=PII(1-P(y=1|x)), which forces P(y=1|x)=PII. Thus Eq. (7) is not an outcome of independent dynamics but a rearrangement of the feedback probability that was already defined as an affine function of the class sum; no new class-sum-to-probability information is added by the derivation.
renaming known result [Section 2.3, Alternative derivation of probability score]
"Since the class sum is clipped at -T and T, you first scale the class sum by T to get values in the range -1 and 1. Then, you shift the score to the range 0 and 2 by adding 1. Finally, you divide that by 2 to get the range of 0 and 1 needed for a valid probability score. Putting all these steps together, you get the equation 1/2(1+v(x)/T), which is the exact same equation as Eq. (7) which was derived by setting TypeI and TypeII feedback probabilities equal each other."

This passage explicitly states that Eq. (7) is fully determined by the clipping bounds of the class sum: any bounded score can be affinely rescaled to [0,1]. Consequently, the 'probability score' is a normalization of v introduced by construction; calling it a probability is a labeling choice rather than an empirical finding. The earlier feedback-balance derivation is therefore redundant for identifying the functional form, and the abstract's claim that the score is 'derived through an analysis of the TM learning dynamics' overstates what the equations establish.

full rationale

The two steps above identify a real but partial circularity. The empirical sections are not circular: Figure 1 tests whether learned class sums track known noise probabilities, and the CIFAR-10 threshold analysis is an external benchmark whose accuracy-vs-confidence result would stand or fall independently of how Eq. (7) is labelled. The paper is also transparent in Section 2.3 that the formula is the natural scaling of a clipped sum. Nevertheless, the central theoretical contribution reduces by construction: Eq. (7) is identical to the definition of PII and to the affine rescaling of v, so its functional form is not independently derived from the TM learning dynamics. The derivation in Section 2.2 further assumes a feedback balance ('there is a point where the probabilities of the two feedback types ... is the same') without deriving it, applies only to positive-polarity clauses, and is contradicted by the paper's own Figure 1 and Section 4 admission that class sums oscillate rather than settle; for negative-polarity clauses the mirror balance gives the opposite equilibrium, so the claimed joint equilibrium cannot exist in a two-polarity TM. These are correctness flaws rather than self-citation or fitted-input circularity, but they leave the 'derived' status of Eq. (7) as mostly definitional, with genuine empirical support confined to the class-sum tracking experiments and the external CIFAR-10 threshold evaluation.

Assumptions & free parameters 2 free parameters · 3 assumptions · 1 invented entities

The central formula depends on the user-set hyperparameter T, on the assumption that class sums are clipped to [-T,T], and on an unproven equilibrium balance between Type I and Type II feedback. The multiclass normalization is introduced without theoretical support.

free parameters (2)

T (target value) = T=2000, 10,000, 20,000 in experiments
The probability score is P=0.5(1+v/T); T is a user-set hyperparameter, not determined by the derivation, so the probability scale is chosen by hand.
s (specificity) = s=1.0, 1.1, 1.2, 2.0, 5.0, 20 in experiments
s controls clause literals and shifts class-sum magnitudes (Figure 2); it is not calibrated for the probability score.

assumptions (3)

ad hoc to paper At equilibrium, the probability of Type I feedback equals the probability of Type II feedback for a clause: P(I|x)=P(II|x).
Invoked in Section 2.2 to solve for v=T(2p-1); no derivation from the TA update dynamics, and class sums visibly oscillate in Figures 1 and 2 rather than settling at a fixed point.
domain assumption The class sum is clipped to [-T,T] for every sample.
Needed so Equation (7) returns values in [0,1]; clipping is asserted in Section 2.1 but weighted class sums from Equation (3) may exceed T, and Figures 1 and 2 show sums at the clip boundary.
ad hoc to paper The normalized multiclass score, defined as the largest individual probability score divided by the sum of scores over all classes, is a valid certainty measure.
Proposed in Section 3.2.2 without theoretical justification; no experiments show that it is calibrated across classes.

invented entities (1)

Normalized probability score (max score divided by sum of scores)
purpose: Convert per-class one-vs-rest scores into a multiclass certainty measure.
No theoretical justification or external calibration is provided; it is a hand-designed normalization.

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Cite this review

Pith. "Pith review of Uncertainty Quantification in the Tsetlin Machine." pith.science (2026). https://pith.science/paper/5CYT4WXD

@misc{pith2026250704175,
  author       = {Pith},
  title        = {Pith review of: Uncertainty Quantification in the Tsetlin Machine},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5CYT4WXD}},
  note         = {Machine review of arXiv:2507.04175}
}

read the original abstract

Data modeling using Tsetlin machines (TMs) is all about building logical rules from the data features. The decisions of the model are based on a combination of these logical rules. Hence, the model is fully transparent and it is possible to get explanations of its predictions. In this paper, we present a probability score for TM predictions and develop new techniques for uncertainty quantification to increase the explainability further. The probability score is an inherent property of any TM variant and is derived through an analysis of the TM learning dynamics. Simulated data is used to show a clear connection between the learned TM probability scores and the underlying probabilities of the data. A visualization of the probability scores also reveals that the TM is less confident in its predictions outside the training data domain, which contrasts the typical extrapolation phenomenon found in Artificial Neural Networks. The paper concludes with an application of the uncertainty quantification techniques on an image classification task using the CIFAR-10 dataset, where they provide new insights and suggest possible improvements to current TM image classification models.

Figures

Figures reproduced from arXiv: 2507.04175 by the authors.

**Figure 2.** Evolution of class sums for the experiment with eight patterns and different [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Prediction certainty by region of TM (a) and ANN (b) models. [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: Visualization of clause counts by region for positive polarity (a) and negative polarity (b) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: Histograms of the individual and normalized probability scores of the CIFAR-10 test set [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Kernel density estimation of max class sum distributions of CIFAR-10 test set predictions. [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: Accuracy and number of samples of subsets of test set samples with different levels of [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: Clause counts of per class of a sample in the test set with label [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: Illustration on CIFAR-10 test images with different number of active clauses. [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [1]

Darshana Abeyrathna, Ole-Christoffer Granmo, Lei Jiao, and Morten Goodwin

K. Darshana Abeyrathna, Ole-Christoffer Granmo, Lei Jiao, and Morten Goodwin. The regression tsetlin machine: A tsetlin machine for continuous output problems. In Paulo Moura Oliveira, Paulo Novais, and Lu \'i s Paulo Reis, editors, Progress in Artificial Intelligence, pages 268--280, Cham, 2019. Springer International Publishing

work page 2019

[2] [2]

Darshana Abeyrathna, Ahmed A

K. Darshana Abeyrathna, Ahmed A. O. Abouzeid, Bimal Bhattarai, Charul Giri, Sondre Glimsdal, Ole-Christoffer Granmo, Lei Jiao, Rupsa Saha, Jivitesh Sharma, Svein A. Tunheim, and Xuan Zhang. Building concise logical patterns by constraining Tsetlin Machine clause size . In IJCAI, 2023

work page 2023

[3] [3]

Massively parallel and asynchronous Tsetlin Machine architecture supporting almost constant-time scaling

Kuruge Darshana Abeyrathna, Bimal Bhattarai, Morten Goodwin, Saeed Rahimi Gorji, Ole-Christoffer Granmo, Lei Jiao, Rupsa Saha, and Rohan K Yadav. Massively parallel and asynchronous Tsetlin Machine architecture supporting almost constant-time scaling . In ICML, 2021

work page 2021

[4] [4]

Tsetlin Machine embedding: Representing words using logical expressions

Bimal Bhattarai, Ole-Christoffer Granmo, Lei Jiao, Rohan Yadav, and Jivitesh Sharma. Tsetlin Machine embedding: Representing words using logical expressions . Findings of EACL, pages 1512--1522, 2024

work page 2024

[5] [5]

The tsetlin machine--a game theoretic bandit driven approach to optimal pattern recognition with propositional logic

Ole-Christoffer Granmo. The tsetlin machine--a game theoretic bandit driven approach to optimal pattern recognition with propositional logic. arXiv preprint arXiv:1804.01508, 2018

arXiv 2018

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The convolutional tsetlin machine

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Smørvik, and Ole-Christoffer Granmo

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Interpretable rule-based architecture for GNSS jamming signal classification

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