The Boltzmann structure of sampling: Intrinsic $p$-value and its emergent closed-form expression

[2608.14608] The Boltzmann structure of sampling: Intrinsic $p$-value and its emergent closed-form expression Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Statistics Theory arXiv:2608.14608 (math) [Submitted on 7 Jul 2026] Title: The Boltzmann structure of sampling: Intrinsic $p$-value and its emergent closed-form expression Authors: Orestis Loukas View a PDF of the paper titled The Boltzmann structure of sampling: Intrinsic $p$-value and its emergent closed-form expression, by Orestis Loukas View PDF HTML (experimental) Abstract: We consider observables $X$ whose realizations in a sampled dataset are restricted, for example by measurement resolution, to a finite set of distinguishable categories within their possibly infinite theoretical domain. Given the probabilities of observable categories, we study the coarse-grained probability mass of families of possible datasets generated through a forward sampling process. The combinatorial construction induces an intrinsically discrete $p$-value defined directly from the sampling process rather than through additional probabilistic structure on observables. Specifying the sample means of $d+k$ arbitrary functions $g_\alpha(X)$ defines a linear family of datasets whose probability mass is obtained as a weighted sum over integer lattice points contained within the associated polyhedron. To overcome the intractable large-$N$ combinatorics, we derive via saddle-point techniques a density approximating these probability masses in the continuum limit of forward sampling within the multinomial universality class. As a demonstration, we consider conditional sampling, where $d$ structural means are fixed while $k$ means vary over admissible datasets. The information geometry emerging from the saddle-point density, together with the spherical symmetry arising at large $N$ from the intrinsic $p$-value construction, enables efficient computation of the $p$-value in the Laplace approximation via the $\chi^2_k$ distribution. The resulting statistic is given by the semi-analytic expression $2N$ times the Kullback-Leibler divergence between the information projections associated with the corresponding structural and observed linear families. These projections can be computed efficiently via standard numerical routines converging for sufficiently well-behaved sample means. Subjects: Statistics Theory (math.ST) ; Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Data Analysis, Statistics and Probability (physics.data-an); Methodology (stat.ME) Cite as: arXiv:2608.14608 [math.ST] (or arXiv:2608.14608v1 [math.ST] for this version) https://doi.org/10.48550/arXiv.2608.14608 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Orestis Loukas [ view email ] [v1] Tue, 7 Jul 2026 00:19:19 UTC (76 KB) Full-text links: Access Paper: View a PDF of the paper titled The Boltzmann structure of sampling: Intrinsic $p$-value and its emergent closed-form expression, by Orestis Loukas View PDF HTML (experimental) TeX Source view license Current browse context: math.ST < prev | next > new | recent | 2026-08 Change to browse by: cond-mat cond-mat.stat-mech math math-ph math.MP physics physics.data-an stat stat.ME stat.TH References & Citations INSPIRE HEP NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... 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