Boltzmann-Informed Probabilities

Yair Neuman; Yochai Cohen

REVIEW 3 major objections 3 minor 7 references

Boltzmann-Informed Probabilities

T0 review · 3 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Boltzmann-informed probabilities, built from odds-derived energy levels, beat raw betting odds in Kelly betting over five English Premier League seasons, with a $1,841.57 cumulative edge.

desk verdict The empirical claim is an artifact of a mis-specified Kelly criterion and an incorrect profit calculation; the underlying idea is not crazy but the paper as written does not support it. read the letter →

arxiv 2505.21543 v1 pith:YR5QXJWH submitted 2025-05-24 physics.soc-ph physics.data-an

Yair Neuman , Yochai Cohen This is my paper · ORCID

classification physics.soc-phphysics.data-an

keywords BoltzmanndistributionKellycriterionsportsbettingEnglishPremierLeagueenergy-informedprobabilitiesDirichletpriormaximumentropyfavorite-longshotbias

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 proposes that assigning hypothetical energy levels to the possible outcomes of a random variable, then feeding those energies through Boltzmann's exponential distribution, can produce better probability estimates than the raw probabilities. It tests this idea on five English Premier League seasons, translating each match's betting odds into energy levels, using the resulting Boltzmann distribution as a prior, and combining it with historical team counts in a Dirichlet posterior update. When those posterior probabilities are used to size bets through the Kelly criterion, they consistently beat bets sized with the original odds, accumulating a $1,841.57 edge over the five seasons. The paper argues that the Boltzmann transform corrects behavioral biases such as the favorite–longshot bias by exponentially sharpening the gap between favorites and underdogs, and that energy-based probability modeling may transfer to other complex social systems.

What carries the argument

The load-bearing device is the mapping from betting odds to energy levels, $E_H=O_H/O_A$, $E_D=O_D$, $E_A=O_A/O_H$, followed by the Boltzmann distribution $p_i=e^{-E_i}/Z$. This transform is what sharpens the odds: it converts a favorite's small implied probability into a tiny Boltzmann weight and an underdog's large implied probability into an even smaller one, so the favorite's posterior probability is amplified. The resulting probabilities are not used as final predictions but as the $\alpha$ parameters of a Dirichlet prior; the posterior is obtained by adding the historical count vector $C=[W_H,D,W_A]$. The evaluation machinery is the fractional Kelly criterion, which decides bet size from the gap between the posterior probability and the odds.

What would settle it

Recompute the five-season Kelly simulation using a valid multinomial likelihood—for each match, use counts of the actual outcomes of previous matches involving the two teams as the Dirichlet observation vector, instead of $C=[W_H,D,W_A]$. If the cumulative edge over the original odds falls materially below $1,841.57 or reverses sign, the reported improvement is an artifact of the invalid count vector rather than a property of Boltzmann-informed probabilities.

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Extended reading notes

Core claim

The central claim is that a probability estimate built from a Boltzmann distribution over odds-derived energy levels is more accurate than the probabilities the odds imply directly. For a match with average odds $O_H,O_D,O_A$, the paper defines $E_H=O_H/O_A$, $E_D=O_D$, $E_A=O_A/O_H$, then sets $p_i=e^{-E_i}/Z$ with $Z=\sum_j e^{-E_j}$. These Boltzmann probabilities are justified as the maximum-entropy distribution under the constraint of the average energy. They are not used as final predictions; they set the $\alpha$ parameters of a Dirichlet prior that is combined with a historical count vector $C=[W_H,D,W_A]$ (where $D$ is the average of the two teams' draw counts) to produce posterior probabilities for the match. Feeding those posterior probabilities into a fractional Kelly criterion with $f=0.2$ and $100 per match yields a cumulative gain of $9,214.74 over five English Premier League seasons, versus $7,373.17 for the same betting rule using the original odds, an edge of $1,841.57. A control that applies the same Bayesian update to the original probabilities earns only $5,222.73, which the paper reads as evidence that the Boltzmann transform, not the Bayesian machinery, produces the improvement.

Load-bearing premise

The load-bearing premise is that the historical counts $C=[W_H,D,W_A]$, where $D$ is the average of the two teams' draw counts, can be treated as a multinomial count vector in a Dirichlet posterior update for the current match; if that premise fails, the posterior probabilities that drive the Kelly simulation are not statistically meaningful and the comparison to raw odds is invalid.

Editorial extensions

If this is right

If the edge is real, any market that publishes odds or prices for mutually exclusive outcomes can be recalibrated with the same parameter-free energy transform, including political and financial prediction markets.
The control result—original odds plus the same Bayesian update earning less than both alternatives—implies that the improvement is attributed to the Boltzmann transform itself, not to the Dirichlet smoothing.
Because the transform amplifies the distance between favorites and underdogs, applying it should counteract the documented favorite–longshot bias in betting markets.
The same energy-based modeling recipe is already pointed toward armed-conflict fatality prediction, suggesting the paper intends the method as a general tool for complex social systems, not just soccer.

Reading between the lines

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

The count vector $C=[W_H,D,W_A]$ does not describe outcomes of a single multinomial trial, so the 'posterior' is not a valid Bayesian posterior for the match; an evaluation that replaces it with counts of actual previous match outcomes would show whether the edge survives a correct likelihood.
The Boltzmann transform is a monotone sharpening of the odds, so the reported edge may be reproducible by any sufficiently strong odds transform; comparing against a fitted softmax with a free temperature would test whether Boltzmann's specific exponential form matters.
The simulation's bet sizing parameters—$f=0.2$ and a $100 per-match bankroll—are arbitrary; sensitivity analyses over these parameters and over bookmaker margins would indicate whether the cumulative edge is robust.

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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

The one genuinely new thing here is using a Boltzmann (softmax) transform on hand-defined energy functions of betting odds, then treating the output as a Dirichlet prior for match outcomes. That is a concrete, testable heuristic and it is honestly presented as an illustration, not as a physics claim. The paper is clearly written and the pseudo-code helps. It does not, however, establish what it claims.

The load-bearing problem is the evaluation. The Kelly fraction is not f*(p - (1-p)/O). That is not the Kelly criterion for decimal odds; it can be positive when the bet has negative expected value. On top of that, a winning bet is credited with stake plus profit instead of profit alone. With decimal odds, a stake S returns profit S*(O-1), not S*O. Both errors inflate returns disproportionately for the Boltzmann probabilities, which push favorite probabilities toward 1 and therefore increase the stake. The reported $1,841.57 edge is very plausibly an accounting artifact rather than evidence of better probability estimates. The stress-test note is right, and it lands exactly on the central comparison in Sections 4.2 and 4.3.

The reader's other main concern also holds: the Dirichlet update treats [WH, (DH+DA)/2, WA] as multinomial counts for a single match. Those counts do not come from one match, they come from two teams' histories, so the "posterior" is not a legitimate Bayesian posterior for the match outcome. That undermines the statistical interpretation even if the Kelly errors were fixed.

What should be credited: the paper does not hide its energy definitions as derived from first principles; it calls them heuristic. It also includes a control (GainOriginal+Bayes) to separate the Boltzmann effect from the Bayesian update. That control is a good instinct, though it inherits the same accounting errors. The max-entropy discussion is fine as motivation but does not justify the specific energy choices, which remain arbitrary. No data or code are provided, so the exact numbers cannot be checked.

Net: the central claim is not credible as written. The idea could be revisited with a correct Kelly formula, a proper predictive model, and significance testing, but this paper does not get there. A serious editor could send it to peer review because the claim is concrete and the flaws are instructive, but I would expect rejection. My own recommendation is to treat it as a desk reject if the journal is selective; the empirical result is an artifact and the method section does not salvage it.

Referee Report

3 major / 3 minor

Summary. The paper proposes 'Boltzmann-informed probabilities' for soccer match outcomes by defining energy levels from betting odds, converting them via the Boltzmann distribution, using the result as a Dirichlet prior over historical win/draw counts, and then evaluating the posterior through a Kelly-criterion betting simulation on five English Premier League seasons. The central claim is that this procedure consistently outperforms probabilities derived directly from the odds, with a cumulative gain edge of $1,841.57.

Significance. If correct, the approach would offer a simple, physics-inspired correction to market-implied probabilities and would be of interest to complex-systems modeling. The manuscript is transparent: it provides pseudocode for every step, uses a concrete five-season dataset, and reports season-by-season results, which makes the claims reproducible in principle. However, the evaluation rests on a mis-specified Kelly criterion, an incorrect profit calculation, and an invalid Dirichlet posterior, so the headline result is not a valid test of probability estimation.

major comments (3)

[§4.2, Pseudocode 2, Step 2] The stake formula KELLY_O = f·(PostO − (1 − PostO)/OddsO) is not the Kelly criterion for decimal odds; the correct fraction is f·(PostO·OddsO − 1)/(OddsO − 1). As written, the formula yields a positive stake whenever PostO > 1/(OddsO + 1), which includes many negative expected-value bets (e.g., OddsO = 2, PostO = 0.4). Consequently, the simulation places bets without a positive edge, and the reported gains cannot be attributed to superior probability estimates.
[§4.3, Pseudocode 2, Step 3] A winning bet is credited with GAIN_O = 100·KELLY_O·OddsO, but with decimal odds the stake is returned along with the winnings, so the profit is 100·KELLY_O·(OddsO − 1). This overstates every winning bet by the stake amount, and since the Boltzmann transform inflates favorite probabilities and hence KELLY_O for favorites, the inflation is systematically larger for the Boltzmann-informed model. The $1,841.57 edge is therefore at least partly an accounting artifact.
[§3.4–§3.5 and Pseudocode 1] The vector C = [WH, D, WA] is treated as a multinomial count vector for the current match in a Dirichlet prior/posterior update, but WH and WA are the historical win totals of the two teams and D is the average of their draw counts; these are not mutually exclusive outcomes of a single exchangeable trial. The resulting posterior is not a valid Bayesian posterior for the match outcome distribution, so the probabilities PostH, PostD, and PostA that drive the simulation are not statistically meaningful.

minor comments (3)

[§3.3] The max-entropy example uses arbitrarily chosen energy values and then asserts that the Boltzmann distribution is the least biased distribution for that average energy; this is true for any energy assignment and does not justify the particular odds-derived energy mapping used in the paper.
[Pseudocode 3] The original odds probabilities PHome, PDrew, and PAway are taken as 1/Odds without normalization, whereas the baseline percentages in §3.1 (32%, 22%, 46%) are normalized; this inconsistency could affect the 'original probabilities' baseline in the Kelly simulation.
[Throughout] There are several typographical issues, including 'PDrew' for 'PDraw', 'Drew' for 'Draw', 'bank role' for 'bankroll', and 'accommodating edge' for 'accumulated edge' in the Results section.

Circularity Check

1 steps flagged · score 4.0 of 10

Max-entropy justification is self-referential; the central Kelly-vs-odds comparison is not circular.

self definitional [Section 3.3, Step 2, max-entropy paragraph (after the Z-normalized pH, pD, pA example)]
"These probabilities maximize entropy for the specific constraint that the average energy equals: E_avg = 0.352∙0.6 + 0.526∙0.2 + 0.122∙1.66 = 0.211 + 0.105 + 0.203 = 0.519. The max entropy principle suggests the least biased inference possible, based on limited information."

The constraint used to invoke the max-entropy principle is not an independent input: E_avg is computed from the Boltzmann probabilities it is supposed to justify. The theorem says that for a given mean energy the least-biased distribution is Boltzmann, but here the mean energy is obtained by plugging the already-computed Boltzmann probabilities back into E_avg = Σ p_i E_i. Thus the justification reduces to 'the Boltzmann distribution is max-entropy for the mean energy that the Boltzmann distribution happens to have,' which is a tautology. The actual choices (EH = OH/OA, ED = OD, EA = OA/OH, and k = T = 1) remain arbitrary ansätze, and the Jaynes-based argument provides no independent constraint that selects them.

full rationale

The paper's central empirical claim—that Boltzmann-informed posterior probabilities yield higher cumulative Kelly gains than raw betting odds—is not circular: the posterior is a well-defined function of the current match's odds and historical counts through match i−1, and the comparison to raw odds is an out-of-sample empirical test with no parameter fitted to the evaluated outcomes. The Boltzmann probabilities themselves are a deterministic softmax-type transform of the same odds used in the baseline, but that alone is not circularity; it simply means the comparison is between two functions of the same input. The main circular element is the max-entropy justification in Section 3.3: the average-energy constraint is computed from the Boltzmann probabilities that it is invoked to justify, so the 'least-biased distribution' argument is self-referential and does not independently motivate the choice of energies or k=T=1. The self-citations [1] and [2] are not load-bearing because the method is fully specified in the paper; the evaluation depends on a nonstandard Kelly/gain calculation, which is a correctness concern rather than a circularity. Overall score reflects one self-definitional justification while the central empirical derivation remains independent.

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

The paper introduces no physical mechanism, only a heuristic mapping from odds to energies. The free parameters are the temperature, the functional forms of the energy levels, and the Kelly fraction. The central statistical assumption, that team-level win counts form a multinomial count vector for a single match, is not justified and is likely false.

free parameters (3)

kT product = 1
Set to k=T=1 in Section 3.3 without justification; controls the sharpness of the exponential transform. No sensitivity analysis is reported.
Energy definitions for match outcomes = EH=OH/OA, EA=OA/OH, ED=OD
Chosen heuristically in Section 3.2; no derivation or robustness check. The draw energy ED=OD has different units from EH and EA, which are ratios.
Kelly fraction f = 0.2
Fractional Kelly fraction in Section 4.2; scales gains for both strategies equally but is an arbitrary choice.

assumptions (3)

ad hoc to paper Boltzmann distribution with kT=1 is a valid probability model for match outcomes.
Invoked in Section 3.3; no physical basis, and kT=1 is set by hand.
domain assumption Maximum entropy principle justifies choosing the Boltzmann distribution.
Cited from Jaynes in Section 3.3, but the energy constraint is arbitrary, so the max-entropy argument is not a valid justification for this specific distribution.
ad hoc to paper Historical win counts of home and away teams and the averaged draw count can be treated as a multinomial count vector for the current match in a Dirichlet posterior.
Used in Sections 3.4 and 3.5 and in the pseudocode; the counts are not mutually exclusive outcomes of a single match, so the conjugate-prior update is statistically invalid.

invented entities (1)

Hypothetical energy levels of match outcomes
purpose: To convert betting odds into a Boltzmann distribution that serves as a Dirichlet prior.
The energies are abstract constructs with no independent falsifiable handle; they are defined only through the odds, and the paper provides no external validation.

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Cite this review

Pith. "Pith review of Boltzmann-Informed Probabilities." pith.science (2026). https://pith.science/paper/YR5QXJWH

@misc{pith2026250521543,
  author       = {Pith},
  title        = {Pith review of: Boltzmann-Informed Probabilities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YR5QXJWH}},
  note         = {Machine review of arXiv:2505.21543}
}

read the original abstract

Traditional interpretations of probability, whether frequentist or subjective, make no reference to the concept of energy. In this paper, we propose that assigning hypothetical energy levels to the outcomes of a random variable can yield improved probability estimates. We apply this Boltzmann-informed approach to the context of sports betting and analyze five seasons of the English Premier League data. It was found that when used to compute the Kelly criterion, Boltzmann-informed probabilities consistently outperform probabilities derived from the original betting odds. These findings demonstrate the value of integrating energy-informed probabilities into studying complex social systems.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

7 extracted references · 6 canonical work pages

[1]

Extreme events in armed conflicts: a perspective

Neuman, Y ., 2025. Extreme events in armed conflicts: a perspective. Frontiers in Physics, 13, p.1528716

work page 2025
[2]

and Cohen, Y ., under review

Neuman, Y . and Cohen, Y ., under review. A minimalist computational model for extreme conflict fatalities

work page
[3]

and Skiera, B., 2009

Spann, M. and Skiera, B., 2009. Sports forecasting: comparing the forecast accuracy of prediction markets, betting odds, and tipsters. Journal of Forecasting, 28(1), pp.55-72

work page 2009
[4]

and De Angelis, L., 2019

Angelini, G. and De Angelis, L., 2019. Efficiency of online football betting markets. International Journal of Forecasting, 35(2), pp.712-721

work page 2019
[5]

Betting Against the Crowd: A Complex Systems Approach

Neuman, Y ., 2024. Betting Against the Crowd: A Complex Systems Approach. New York: Springer

work page 2024
[6]

and Whelan, K., 2025

Hegarty, T. and Whelan, K., 2025. Forecasting soccer matches with betting odds: A tale of two markets. International Journal of Forecasting, 41(2), pp.803-820

work page 2025
[7]

Information theory and statistical mechanics

Jaynes, E.T., 1957. Information theory and statistical mechanics. Physical Review, 106(4), pp.620–630. https://doi.org/10.1103/PhysRev.106.620 15 Appendix 1. PSEUDOCODE 1. Generating the Boltzmann-informed probabilities For each match i = 1 to N: # Step 1: Convert betting odds to probabilities PHome[i] = 1 / OddsHome[i] PDrew[i] = 1 / OddsDrew[i] PAway[i]...

doi:10.1103/physrev.106.620 1957

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Extreme events in armed conflicts: a perspective

Neuman, Y ., 2025. Extreme events in armed conflicts: a perspective. Frontiers in Physics, 13, p.1528716

work page 2025

[2] [2]

and Cohen, Y ., under review

Neuman, Y . and Cohen, Y ., under review. A minimalist computational model for extreme conflict fatalities

work page

[3] [3]

and Skiera, B., 2009

Spann, M. and Skiera, B., 2009. Sports forecasting: comparing the forecast accuracy of prediction markets, betting odds, and tipsters. Journal of Forecasting, 28(1), pp.55-72

work page 2009

[4] [4]

and De Angelis, L., 2019

Angelini, G. and De Angelis, L., 2019. Efficiency of online football betting markets. International Journal of Forecasting, 35(2), pp.712-721

work page 2019

[5] [5]

Betting Against the Crowd: A Complex Systems Approach

Neuman, Y ., 2024. Betting Against the Crowd: A Complex Systems Approach. New York: Springer

work page 2024

[6] [6]

and Whelan, K., 2025

Hegarty, T. and Whelan, K., 2025. Forecasting soccer matches with betting odds: A tale of two markets. International Journal of Forecasting, 41(2), pp.803-820

work page 2025

[7] [7]

Information theory and statistical mechanics

Jaynes, E.T., 1957. Information theory and statistical mechanics. Physical Review, 106(4), pp.620–630. https://doi.org/10.1103/PhysRev.106.620 15 Appendix 1. PSEUDOCODE 1. Generating the Boltzmann-informed probabilities For each match i = 1 to N: # Step 1: Convert betting odds to probabilities PHome[i] = 1 / OddsHome[i] PDrew[i] = 1 / OddsDrew[i] PAway[i]...

doi:10.1103/physrev.106.620 1957