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REVIEW 4 major objections 5 minor 3 references

Random Variables aren't Random

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

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Pith's one-line read Random variables are measurable functions, and statistical inference can proceed by pure logic without hypothetical repeated sampling.

desk verdict A readable restatement of Fisher's disjunction with the technical load carried by the author's prior papers; the novelty is packaging, not proof. read the letter →

arxiv 2502.06628 v1 pith:XSMSI5YO submitted 2025-02-10 stat.OT

Paul W. Vos This is my paper · ORCID
classification stat.OT MSC 62A0162B1062F0362F25
keywords randomvariablesmeasurablefunctionsstatisticalinferencereductioadabsurdumtailareasinformationtheorybias-variancetradeofffrequentistfoundations
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

The paper argues that randomness is not a property of the mathematical objects statisticians call random variables; they are measurable functions, and their distributions are fully specified mathematical structures. It claims that defining and using these distributions requires no appeal to infinite hypothetical samples, and that the only real randomization involved is the single physical mechanism that produced the observed sample. From this, statistical inference is recast as a logical argument: if the observed data sit in the extreme tail of a hypothesized distribution, the conclusion is that either the hypothesis is false or a rare event has occurred. The paper then replaces bias and variance as criteria for judging estimators with information-based measures that depend only on the probabilities assigned to possible data, not on the units or ordering of the data. If these claims hold, long-standing disputes over repeated-sampling interpretations of p-values and confidence intervals dissolve into a purely mathematical foundation.

What carries the argument

The central mechanism is the Fisher-Information-Logic (FIL) approach: a modified reductio ad absurdum in which the probability that the observed sample falls in a tail of a hypothesized distribution supplies the contradiction. The objects doing the work are simple distributions—frequencies normalized on a finite label space—extended to discrete and continuous distributions by measure theory, and tail areas computed from them. 'Rare' is defined by a tail-area threshold α, which enters through an auxiliary hypothesis Haux, the statement that the observed sample is not exceptionally rare; the argument then partitions the model family into the set Mα where no contradiction arises and its complement where it does. These tail-based contradictions yield confidence regions, while generalized estimators and their information content replace bias and variance as the tools for comparing statistical procedures.

What would settle it

Take one dataset and one null model, and compute the paper's reductio for two thresholds, say tail area 0.05 and 0.001; if one threshold yields 'hypothesis false or rare event' and the other yields no contradiction for the same observation, the logical verdict depends on the analyst's chosen α rather than on mathematics alone. Alternatively, exhibit two physical sampling plans that could both have produced the same observed sample but yield different combinatorial distributions, showing that a single randomization does not by itself fix the sampling distribution.

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Extended reading notes

Core claim

The paper's central claim is that random variables, properly understood as measurable functions, are fully characterized by their associated distributions on label spaces, and that this characterization needs no reference to randomization or repeated sampling. It develops a hierarchy of distributions—simple distributions on finite label spaces, then discrete and continuous generalizations via measure theory—and shows how percentiles and tail areas locate an observed sample within a sampling distribution. From this it extends the reductio ad absurdum of classical deduction to induction: given a null hypothesis and an auxiliary hypothesis that the observed value is not exceptionally rare, a value in the extreme tail produces the logical conclusion 'either the hypothesis is false or a rare event has occurred.' The discovery is that this logical scaffolding, together with single-instance physical randomization, supports confidence regions and point estimation without hypothetical infinite samples, and that estimator assessment should be based on the information content of the probability assignments rather than on bias and variance defined through label-space structure.

Load-bearing premise

The load-bearing premise is that one physical randomization is enough to connect the observed sample to its mathematical sampling distribution, together with the premise that 'exceptionally rare' has an objective, threshold-independent meaning as a tail area; if either premise fails, the induced disjunction is just an ordinary significance test.

Editorial extensions

If this is right

  • P-values become statements about where an observed sample sits in a fixed sampling distribution, so their justification no longer depends on what would happen across repeated datasets.
  • A single physical randomization—the one that produced the data—is enough to connect the observed sample to the mathematical sampling distribution; infinite hypothetical samples are unnecessary.
  • Confidence regions inherit the logical reading that either the true model lies in the region or the observed sample is in the tail of every excluded model's sampling distribution.
  • Estimator performance is evaluated by the information content of probability assignments, so conclusions do not change when the label space is re-parameterized, as with km/L versus L/km units.
  • In exponential families, point estimates can be treated as distributions in the model family itself, generalizing least squares through Kullback-Leibler divergence.

Reading between the lines

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

  • A natural extension is to re-read established frequentist procedures—likelihood ratios, p-values, confidence sets—as logical location statements in model space; the formulas would survive while the repeated-sampling story would be optional.
  • One testable extension would compare orderings of estimators by information content against finite-sample prediction error in cases where bias and MSE disagree, to see whether the information criterion tracks practical performance.
  • The framework's attribution of randomness to physical sampling mechanisms alone could sharpen the justification of randomized experiments in causal inference, where the design feature that matters is the single realized randomization, not hypothetical re-randomizations.
  • Because the auxiliary rarity threshold α remains a choice, the purely logical status of induction could be tested by asking whether a principled, for example minimax or decision-theoretic, choice of α removes the residue of convention.
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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 random variables should be understood as measurable functions and that their distributions can be developed through measure-theoretic proportions without invoking infinite hypothetical repeated sampling. It uses Penrose's three-world distinction to locate randomization in the physical world and sampling distributions in the mathematical world. The paper then proposes an 'inductive reductio' based on an auxiliary hypothesis Haux that the observed value is not exceptionally rare, yielding Fisher's disjunction that either the hypothesis is false or a rare event occurred. Section 7 introduces generalized estimators and an information-based measure Lambda(g) that is claimed to replace bias and variance as assessment tools. The three stated contributions are: a characterization of random variables without hypothetical samples, an extension of deduction to induction, and an information-based assessment of statistical procedures.

Significance. If the framework is correct, it would provide a clean conceptual separation between mathematical distributions and physical sampling, a logical reading of p-values, and a label-space-invariant way to compare estimators. The paper is clearly written, the lottery example is worked carefully, and the discussion engages seriously with Fisher's actual views and with the p-value debate. Its strengths include a worked two-lottery example and an explicit link to information geometry. However, the technical core of the third contribution is deferred almost entirely to self-cited earlier work, and the inductive reductio depends on an analyst-chosen rarity threshold, so the significance as a standalone contribution is conditional on those gaps being resolved.

major comments (4)
  1. [5.2 and 7.2] The formalization of Haux is not objective. Section 5.2 first labels a sum of 28 as rare and 27 as not rare, corresponding to a tail area of 0.0002 for lottery A, and then labels a sum of 24 as extreme and 23 as not, corresponding to a tail area of 0.0171. No principle in the paper selects one tail-area cutoff over the other. With Haux formalized in Section 7.2 as TA(yobs, m) > alpha, the reductio conclusion not(H0 and Haux) is exactly Fisher's disjunction 'either the model is false or the observation fell in an alpha-tail,' and the set M_alpha in Section 7.2 is precisely the acceptance region of a level-alpha test. Since both the test statistic (the ordering) and alpha are analyst choices, the claimed objective extension of deduction to induction is not established; unless Haux is derived from H0 plus logic alone, the argument restates significance testing with a free alpha.
  2. [7.3 and 8] The central technical apparatus for the third contribution is not present in this manuscript. The definition of the information content Lambda(g), the generalized estimators g(y, m), and the claim in Section 8 that for an unbiased estimator Lambda(theta-hat) equals the reciprocal of its variance are all deferred to Vos (2022), Vos and Wu (2024), and Vos (2024). No theorem or proof of these properties appears here, so the central claim that information replaces bias and variance cannot be independently evaluated from this paper. The manuscript should either state the definitions and derivations or explicitly mark these as review of prior work with the relevant results reproduced.
  3. [5.2] The logical step from H0 and Haux to the conclusion sobs in S27 presupposes the rarity ordering and threshold rather than deriving them. Haux as introduced is the statement 'the observed sum is not exceptionally rare,' which is a property of the observed value relative to a tail-area cutoff; by itself it does not place sobs in the set S27 unless one already assumes that 28 is rare and that the cutoff is the 99.98th percentile. The reductio therefore does not supply an objective bridge from deductive to inductive logic; it assumes the bridge in the formulation of Haux.
  4. [2 and Abstract] The first contribution is not backed by a formal statement in the body. The abstract claims that 'random variables, properly understood as measurable functions, can be fully characterized without appealing to infinite hypothetical samples,' but Section 2 defines distributions (simple, discrete, and continuous) and never states a formal definition of a random variable as a measurable function or proves a characterization theorem. Without such a definition, the first contribution is a philosophical restatement rather than a demonstrated mathematical result.
minor comments (5)
  1. [2.1] The phrase 'order pair' should be 'ordered pair' in the description of the multi-set representation.
  2. [References] In the reference for Vos (2024), 'Etimators' should be 'Estimators'.
  3. [2.2] The notation XK is used both for the label space and for the distribution on that space (e.g., 'Pr(XK = x) = mK(x)'), which is confusing; a separate symbol for the distribution would clarify the exposition.
  4. [5.3] The comment that the likelihood ratio is constant across the outcomes {0,1,...,7} in the lottery example is not explained; a brief derivation or an explanatory sentence would help readers see why this matters for the ordering.
  5. [7.2] The symbol M_alpha^complement is used without defining the complement operation; the text should state explicitly that the complement is taken within the model family M.

Circularity Check

2 steps flagged · score 8.0 of 10

Two of the three key contributions reduce to inputs: the 'logical induction' is the negation of the self-defined Haux, and the information-based assessment is deferred to the author's own prior papers.

  1. self definitional [Section 5.2, 'Induction and Logic'; restated in Section 7.2]
    "This extension introduces an auxiliary hypothesis Haux: “the observed sum is not exceptionally rare.” ... we can express Haux as the statement that sobs falls below the 99.98th percentile ... This formalization allows us to construct a reductio ad absurdum argument applicable for induction: ... ∴ ¬(H◦ ∧ Haux) ≡ ¬H◦ ∨ ¬Haux ... The conclusion takes the form of a logical disjunction that Fisher (1960) described as: Either the hypothesis is not true, or an exceptionally rare outcome has occurred."

    The 'inductive' conclusion is the negation of the conjunction whose second conjunct is Haux. Since Haux is defined as 'not exceptionally rare', ¬Haux is 'exceptionally rare', so the derived disjunction '¬H0 ∨ rare outcome' is exactly the De Morgan expansion of ¬(H0 ∧ Haux). The result is therefore contained in the definition of Haux plus modus tollens; no new inductive principle is supplied. The threshold is also analyst-calibrated, as the paper itself gives 0.0002 and 0.0171 as alternative cutoffs, so the claimed 'objective' logical inference reduces to a tail-area significance statement.

  2. self citation load bearing [Sections 7.1 and 7.3]
    "Technical details of generalized estimation are given in Vos (2022) and Vos & Wu (2024). See Vos & Holbert (2022) for additional discussion regarding the adequacy of a single random sample for inference. ... the effectiveness of a generalized estimator is measured through its information content Λ(g)."

    The paper's third contribution—information-based assessment replacing bias and variance—is not derived here. The central objects, generalized estimators and their information content Λ(g), are never defined in this manuscript; their properties are delegated to three self-cited papers by the same author. The single-randomization adequacy premise is likewise supported only by Vos & Holbert (2022). These self-citations are load-bearing because the FIL framework and its information metrics would be uninterpretable without them, and no formal proof or independent reproduction is provided.

full rationale

The first contribution—random variables as measurable functions without repeated sampling—is standard measure-theoretic content and is not circular. The second contribution, however, is a valid modus tollens whose conclusion is the negation of the auxiliary hypothesis Haux; because Haux is defined as 'the observed sum is not exceptionally rare', the 'Fisher disjunction' is just the De Morgan equivalent of the starting conjunction, with the rarity threshold chosen by the analyst (0.0002 vs 0.0171). The third contribution is not established in this manuscript: generalized estimators and Λ(g) are named but undefined, and the paper explicitly sends the reader to Vos (2022), Vos & Wu (2024), and Vos & Holbert (2022) for the technical content and for the single-randomization justification. These self-citations carry the central information-based assessment, so two of the three stated contributions reduce to a self-citation chain or to a definition. The paper contains independent pedagogical content about measure-theoretic distributions, which keeps the score below 10, but the advertised advances are substantially circular as presented.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The framework depends on several unproven philosophical premises and on technical machinery containing no shipped code or formal proofs. The ledger shows that the paper's contribution is mostly a reassembly of standard measure theory plus self-cited prior work, with the analyst-chosen rarity threshold and test statistic doing much of the work.

free parameters (2)
  • Significance level alpha for defining rare outcomes = user-selected, e.g., 0.05
    Haux is formalized as TA(yobs,m) > alpha in Section 7.2; different alpha values change which models are rejected and the confidence region Malpha.
  • Test statistic or generalized estimator g = analyst-chosen
    Section 5.3 states the ordering is an important choice; Section 7.2 defines tail areas through g(y,m), so conclusions depend on the chosen ordering.
assumptions (5)
  • domain assumption A real-world population can be exactly modeled by a simple distribution with finite frequencies.
    Section 2.1 defines simple distributions and Section 3 says X(N,K) provides an exact model for the real world population.
  • domain assumption Single physical randomization of the observed sample is sufficient to connect the sample to the mathematical sampling distribution.
    Section 1 says this requires only the single randomization that produced the data; Section 7.1 repeats the point.
  • ad hoc to paper Penrose's three worlds (physical, mental, mathematical) are a valid and useful ontology for statistics.
    Section 3 invokes Penrose (2007); the entire argument separating random variables from randomness rests on this classification.
  • ad hoc to paper Haux, that the observed sample is not exceptionally rare, can be treated as a formal hypothesis in a reductio argument.
    Section 5.2 introduces this auxiliary hypothesis to extend deduction to induction; it is not derived from measure theory.
  • ad hoc to paper Probability assignments, not label-space structure, are the right basis for comparing estimators.
    Section 6 concludes that one should ignore the structure of the label space and use probability or density to assess estimators; this normative premise drives the information-based assessment.

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Cite this review

Pith. "Pith review of Random Variables aren't Random." pith.science (2026). https://pith.science/paper/XSMSI5YO

@misc{pith2026250206628,
  author       = {Pith},
  title        = {Pith review of: Random Variables aren't Random},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XSMSI5YO}},
  note         = {Machine review of arXiv:2502.06628}
}
read the original abstract

This paper examines the foundational concept of random variables in probability theory and statistical inference, demonstrating that their mathematical definition requires no reference to randomization or hypothetical repeated sampling. We show how measure-theoretic probability provides a framework for modeling populations through distributions, leading to three key contributions. First, we establish that random variables, properly understood as measurable functions, can be fully characterized without appealing to infinite hypothetical samples. Second, we demonstrate how this perspective enables statistical inference through logical rather than probabilistic reasoning, extending the reductio ad absurdum argument from deductive to inductive inference. Third, we show how this framework naturally leads to information-based assessment of statistical procedures, replacing traditional inference metrics that emphasize bias and variance with information-based approaches that better describe the families of distributions used in parametric inference. This reformulation addresses long-standing debates in statistical inference while providing a more coherent theoretical foundation. Our approach offers an alternative to traditional frequentist inference that maintains mathematical rigor while avoiding the philosophical complications inherent in repeated sampling interpretations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

3 extracted references · 2 canonical work pages

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    Efron, B. (2024). Machine learning and the James–Stein estimator. Jpn. J. Stat. Data Sci. , 7(1), 257–266. 16 Fisher, R. (1959). Statistical methods and scientific inference . Hopetoun Street, University of Edinburgh: T and A Constable Ltd, 2nd edition. Fisher, S. R. A. (1960). Scientific thought and the refinement of human reason- ing. 3, 1–10. Gelman, A...

    work page 2024

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    Good, I. J. (1971). Letters to the editor: 46656 varieties of bayesians. The American Statistician, 25(5), 62–63. Penrose, R. (2007). The road to reality: a complete guide to the laws of the universe. New York: Vintage Books, 1st vintage books ed edition. Spiegelhalter, D. (2024). Why probability probably doesn’t exist (but it is useful to act like it doe...

    work page 1971

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    Vos, P. (2022). Generalized estimators, slope, efficiency, and fisher information bounds. Information Geometry . SharedIt link https://rdcu.be/c0YQn. Vos, P. (2024). Rethinking Mean Square Error: Why Information is a Superior Assessment of Etimators. Preprint, https://arxiv.org/abs/2412.08475. Vos, P. & Holbert, D. (2022). Frequentist statistical inferenc...

    arXiv 2022

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Reviewed August 8, 2026 · model on record in the stance chip above.