A framework for general discrete probability calculations

Kacper Topolnicki; Roman Skibi\'nski

REVIEW 2 major objections 4 minor 14 references

A framework for general discrete probability calculations

T0 review · 2 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The paper claims that any discrete probability query reduces to one linear system, with maximum entropy supplying missing constraints.

desk verdict A clean, reproducible software-description paper whose central claim overreaches: the entropy-maximization completion breaks on natural zero-probability inputs, a gap the authors never flag. read the letter →

arxiv 2607.21011 v1 pith:KYG4SBB5 submitted 2026-07-23 physics.data-an

Kacper Topolnicki , Roman Skibiński This is my paper · ORCID

classification physics.data-an

keywords probabilitylogicdisjunctivenormalformconditionalentropymaximizationdiscretedistributionsBayeslinearsystems

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

Probability questions built from logical statements can be answered mechanically. The paper sets out to show that if a user specifies conditional probabilities in the form P(a|b)=α, each statement becomes one row of a linear system whose variables are the probabilities of all elementary conjunctions of the given literals. Solving that system gives every atom probability, and any desired conditional probability is then a ratio of two linear combinations. When the equations do not determine a unique distribution, the framework selects the maximum-entropy completion, so a single answer is produced rather than a family.

What carries the argument

The key object is the probability vector p indexed by the full disjunctive-normal-form atoms of the literals. Every user statement is a single row of a linear system in p; every query is a ratio of linear forms in p. For non-unique systems, the null-space form M y=0, an interior positive vector y(+) found by linear programming, and a parametrization of all positive solutions define the search space over which Shannon entropy is maximized.

What would settle it

Feed the framework a single literal a with the statement P(a|true)=0. The entropy step requires an all-positive solution to the null-space system, but every feasible vector has zero on the atom containing a, so the linear program cannot return a starting vector. If the implementation does not provide any probability for a query in this case, the general claim 'any probability related to the system' is refuted for that input.

Watch

Extended reading notes

Core claim

The framework's central move is to write every logical statement in full disjunctive normal form: a fixed set of literals S generates 2^|S| mutually exclusive atoms, and any proposition is a disjoint union of atoms. Each user-supplied statement P(u|v)=α is transformed via the product rule into one linear equation in the atom probabilities, and the whole set of statements becomes an augmented matrix (A|b). A solution vector p lists the atom probabilities, and equation (17) expresses any query P(u|v) as a ratio of sums over p. When the system is underdetermined, the problem is recast as a null-space equation M y=0; a linear program finds an all-positive vector y(+), which seeds a parameterized

Load-bearing premise

The entropy-maximising procedure assumes that the set of feasible probability vectors contains one with all components strictly positive; if a user statement or logical constraint forces any elementary-conjunction probability to zero, the linear program that finds the starting vector has no feasible solution.

Editorial extensions

If this is right

Any discrete distribution whose state is described by a finite set of logical literals can be queried for arbitrary conditional probabilities once the constraints are entered as linear equations.
When the constraints are insufficient, the maximum-entropy completion provides a unique, reproducible answer.
User-declared logical constraints can shrink the DNF atom set, reducing the size of the linear system and making larger problems tractable.
The same linear machinery can be embedded in other software because the matrices are exposed and the implementation uses only common numerical and symbolic libraries.
The worked three-sample experiment demonstrates that the framework reproduces the counterintuitive 1/3 versus 2/3 probabilities without any special-case reasoning.

Reading between the lines

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

The paper leaves implicit that if a constraint forces one atom probability to zero, the entropy step's requirement of an all-positive starting vector fails; a workable extension would need to handle degenerate feasible sets, for instance by dropping forced-zero atoms before the linear program.
The exponential growth of the DNF basis limits direct application to small literal counts; combining the constraint pruning with factored or sampled representations could extend the approach to larger realistic systems.
A natural testable extension is to compare the entropy completion against known closed-form maximum-entropy solutions for small underdetermined systems, verifying that the gradient-based optimizer reaches the true maximizer.
The framework could serve as a probabilistic engine for natural-language probability checks, as the paper hints, by turning annotated text statements into logical constraints.

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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. First, this is a competent, honest software-description paper: the DNF-to-linear-equation translation is standard but cleanly worked, the code is real and shipped with examples and tests, and the Monty Hall walkthrough is correct and pedagogical. Second, the abstract's claim that the framework lets you calculate "any probability" is too strong. The stress-test note is right: Section 5's entropy-maximization procedure requires a strictly positive null vector y^+ with all components >1. If a user inputs P(a|true)=0, every DNF atom containing a must be zero, the feasible set sits on the boundary of the simplex, and the LP has no positive starting point. The paper never discusses this case, and logical constraints don't cover it because P=0 is a probability statement, not a constraint of the form c=true. So exactly in the underdetermined regime where maxent is supposed to operate, a common input class silently fails. That's a completeness gap in the central claim, not a numerical quibble.

What's genuinely useful: the parametrization trick in Section 5 (finding a positive solution and then exploring bounded perturbations via unscaled parameters) is a reasonable engineering choice, and the library's design—exposing matrices, relying only on numpy/torch/sympy/scipy—makes it easy to embed. The authors also openly acknowledge the exponential blowup with number of literals, which is honest. The code is the strongest part; the examples in Section 6 reproduce quickly.

Soft spots beyond the positivity issue: novelty is modest. The math is a restatement of Cox/Jaynes plus standard maxent; the only new content is the software and the parametrization. The literature review is shallow—claiming a lack of discrete probability software ignores a lot of probabilistic programming work. The linter is mentioned but not described, so the "natural language" promise hangs on a reference to a separate repository.

Who this is for: someone who wants a lightweight tool for small discrete probability systems with well-constrained, strictly positive solutions. It's not a conceptual advance. It deserves peer review—the code is reproducible and the issue I named is fixable—but the revision should either handle boundary cases (e.g., by splitting zero-probability atoms out of the system before maxent) or explicitly restrict the claimed scope. I'd send it to a referee, and I'd tell them to probe the zero-probability case first.

Referee Report

2 major / 4 minor

Summary. The paper proposes a framework for discrete probability calculations. User-specified conditional probability statements of the form P(u|vx)=α are converted into rows of a linear system whose unknowns are the probabilities of full DNF atoms (eqs. 14, 16). When the linear system is underdetermined, the framework selects a solution by maximizing entropy: it first uses a linear program to find a strictly positive point y^(+) in the nullspace of M=(A|-b), then optimizes over the remaining degrees of freedom while preserving positivity. The authors provide a public Python library (ProbabilityIter) and demonstrate the method on a Monty Hall variant and on an entropy-maximization example.

Significance. The translation of probability statements into linear equations is mathematically straightforward and appears correct for consistent inputs, and the public, lightweight implementation is a practical contribution. The Monty Hall example is reproduced correctly. However, the central claim that the framework can calculate 'any probability' is compromised by two gaps in the underdetermined case: the strict-positive-point requirement fails for natural boundary solutions, and the entropy-maximization procedure as described does not enforce normalization. Both issues are load-bearing for the main novel part of the paper (Section 5). If resolved, the framework could be a useful tool for discrete Bayesian reasoning and natural-language probability checking.

major comments (2)

[Section 5, LP setup] The method requires y^(+) with all coordinates strictly greater than 1 and satisfying M~ y^(+)=0. This interior-point assumption fails whenever the input forces any DNF atom probability to zero. For example, P(a|true)=0 makes every atom containing a have zero mass, so the feasible set lies on the boundary of the simplex and the LP is infeasible. Section 4.1's DNF_c reduction only covers logical constraints c, not zero-probability statements, so this case is not handled. The central claim to calculate 'any probability' is therefore not implemented for a natural class of underdetermined inputs. Please either handle boundary constraints by atomic reduction or by using a nonnegative/barrier method, and state the restriction.
[Section 5 and Section 6.2] The entropy objective S=-Σ p_i ln p_i is maximized over the affine set Ap=b, but no equation enforces the normalization Σ p_i=1. For the Section 6.2 example with only two P(atom|true)=0.02 constraints, the feasible p vectors have arbitrary total mass; without a sum-to-1 constraint, the maximum over the positive orthant would place the remaining atoms near e^{-1}≈0.37 each, not at the plotted plateau ≈0.0038. The plotted result matches a maximum that enforces normalization, but such a constraint is not described. If the implementation adds P(true|x)=1 as a default row, this must be stated; otherwise the described procedure is not well-posed for underdetermined systems.

minor comments (4)

[Throughout] Numerous spelling errors: 'deriviations', 'purpouses', 'respecively', 'transoformation', 'satistied', 'Sollition', 'schould', 'succesfull', 'tward', 'plateou', 'desribed'. A thorough proofread is needed.
[Section 5, code listing] The method name 'getPositiveSollition' in the example code is a typo; should be 'getPositiveSolution'. Also 'transoformation' in the text.
[Reference [10]] The LP construction is attributed to a Mathematics Stack Exchange post. For a journal article, please replace this with a textbook or standard reference on linear programming or convex geometry.
[Figure 3] The axis labels and caption contain typos ('plateou', 'tward'), and the y-axis range makes the small plateau values difficult to read. Consider adding the sum of displayed probabilities in the caption.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the probability-vector construction is self-contained, and the entropy completion is an explicitly declared modeling choice rather than a disguised fit.

full rationale

The derivation chain is self-contained: each user statement P(u|vx)=α is expanded, via the sum and product rules (2)-(3), into a single row of Ap=b using equations (14) and (16), and the target probability P(u|vx) is then evaluated from the solution vector p using equation (17). The Monty Hall demonstration (Subsection 6.1) is fully determined by the logical constraints and the symmetry assumptions stated in lines 7-13; the outputs 1/3 and 2/3 are derived from those constraints, not fitted to a target quantity, and they match an external benchmark. Section 5's entropy maximization is explicitly introduced as a completion rule for underdetermined systems ('In case there are too few statements to provide a unique answer, entropy maximization is used to fill in the missing information'), so the maximum-entropy output is by construction the result of that declared rule, not an unannounced circular prediction. The self-citation [1] refers to the accompanying software implementation and is not used to justify any theorem; the other citations are standard external sources. The positive-null-vector requirement in Section 5 (y^(+)_j > 1) is a real completeness/correctness limitation for zero-probability or boundary constraints, but it is not a circularity: the framework's linear-system reduction does not presuppose the entropy result. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors is imported, and no ansatz is smuggled in via self-citation. Therefore the paper merits a circularity score of 0.

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

The paper adds no new free parameters or entities; it applies known probability rules and maxent. The key additional assumptions are the maxent completion rule and the stricter positivity requirement used by the optimization, the latter being an impractical restriction for zero-probability scenarios.

assumptions (4)

standard math Probability rules of sum and product (Cox/Jaynes) apply to logical propositions.
Used throughout Section 2, equations (2) and (3).
standard math Every logical statement v over literals S can be expressed in full DNF and the DNF atoms are mutually exclusive and exhaustive.
Section 3, equation (5).
domain assumption When user constraints are underdetermining, the maximum-entropy distribution is the canonical completion.
Section 5; this is a standard maxent principle but not derived in the paper.
ad hoc to paper The linear system Ap = α b has a solution with all components strictly positive (y^(+) with y_j > 1).
Section 5; required by the linear-programming step, but not guaranteed when constraints force some atom probabilities to zero.

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Cite this review

Pith. "Pith review of A framework for general discrete probability calculations." pith.science (2026). https://pith.science/paper/KYG4SBB5

@misc{pith2026260721011,
  author       = {Pith},
  title        = {Pith review of: A framework for general discrete probability calculations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KYG4SBB5}},
  note         = {Machine review of arXiv:2607.21011}
}

read the original abstract

Probability follows a simple and concise set of rules. In practice, however, reason- ing about probability may be highly unintuitive and this leads to the possibility of miscalculations even for simple problems. We present an approach to facilitate the correct description of systems governed by discrete probability. The methods described in this paper allow making general statements about the values of prob- ability. This information can next be used to calculate any probability related to the system described by the statements. In case there are too few statements to provide a unique answer, entropy maximization is used to fill in the miss- ing information. The approach has wide potential applications that range from experimental physics to natural language analysis. The theoretical framework described here is also the basis of a software implementation that is designed to be straightforward to extend and requires only popular python libraries, numpy, torch, sympy, and scipy.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

14 extracted references · 2 canonical work pages

[1]

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Topolnicki, K.: ProbabilityIter. GitHub (2026). https://github.com/ kacpertopolnicki/ProbabilityIter

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Abril-Pla, O., Andreani, V., Carroll, C., Dong, L., Fonnesbeck, C.J., Kochurov, M., Kumar, R., Lao, J., Luhmann, C.C., Martin, O.A., Osthege, M., Vieira, R., Wiecki, T., Zinkov, R.: PyMC: A modern and comprehensive probabilistic programming framework in Python. PeerJ Computer Science9(e1516) (2023) https://doi.org/10.7717/peerj-cs.1516

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Sikora, K.: ProbabilityLinter. GitHub (2026). https://github.com/ krystian-sikora/ProbabilityLinter 14

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

GitHub (2026)

Topolnicki, K.: ProbabilityIter. GitHub (2026). https://github.com/ kacpertopolnicki/ProbabilityIter

2026

[2] [2]

PeerJ Computer Science9(e1516) (2023) https://doi.org/10.7717/peerj-cs.1516

Abril-Pla, O., Andreani, V., Carroll, C., Dong, L., Fonnesbeck, C.J., Kochurov, M., Kumar, R., Lao, J., Luhmann, C.C., Martin, O.A., Osthege, M., Vieira, R., Wiecki, T., Zinkov, R.: PyMC: A modern and comprehensive probabilistic programming framework in Python. PeerJ Computer Science9(e1516) (2023) https://doi.org/10.7717/peerj-cs.1516

doi:10.7717/peerj-cs.1516 2023

[3] [3]

Journal of Open Source Software4(34), 1265 (2019) https://doi.org/10.21105/joss.01265

Harpole, A., Zingale, M., Hawke, I., Chegini, T.: pyro: a framework for hydro- dynamics explorations and prototyping. Journal of Open Source Software4(34), 1265 (2019) https://doi.org/10.21105/joss.01265

work page doi:10.21105/joss.01265 2019

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GitHub (2026)

Sikora, K.: ProbabilityLinter. GitHub (2026). https://github.com/ krystian-sikora/ProbabilityLinter 14

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Johns Hopkins Univer- sity Press, ??? (2001)

Cox, R.: Algebra of Probable Inference. Johns Hopkins Univer- sity Press, ??? (2001). https://doi.org/10.56021/9780801869822 . http://dx.doi.org/10.56021/9780801869822

doi:10.56021/9780801869822 2001

[6] [6]

Cambridge University Press, ??? (2003)

Jaynes, E.T.: Probability Theory: The Logic of Science. Cambridge University Press, ??? (2003). https://doi.org/10.1017/cbo9780511790423 . http://dx.doi.org/10.1017/CBO9780511790423

doi:10.1017/cbo9780511790423 2003

[7] [7]

(ed.) From Rationality and Consistency to Bayesian Probability, pp

Smith, C.R., Erickson, G.: In: Skilling, J. (ed.) From Rationality and Consistency to Bayesian Probability, pp. 29–44. Springer, Dordrecht (1989). https://doi.org/ 10.1007/978-94-015-7860-8 2 .https:doi.org10.1007978-94-015-7860-8 2

work page doi:10.1007/978-94-015-7860-8 1989

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Dover Publications, New York, NY (2018)

Kolmogorov, A.N.: Foundations of the Theory of Probability. Dover Publications, New York, NY (2018)

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