measure theory

Existing formulations of the fundamental theorem of agnostic PAC learning implicitly assume measurability conditions without explicitly specifying the minimal requirements for its rigorous validity. Method: Grounded in measure theory, this work systematically identifies and makes explicit the minimal measurability assumptions necessary for the theorem’s strict validity. It integrates VC theory, model theory—particularly NIP and o-minimal structures—and statistical learning theory to deliver, for the first time, a self-contained, measure-theoretically rigorous proof. Contributions: (1) A precise statement and fully rigorous proof framework for the foundational theorem; (2) Sufficient conditions for PAC learnability of hypothesis classes over o-minimal structures; (3) A proof that binary-classification neural networks with standard activation functions—including ReLU and sigmoid—are PAC learnable under o-minimal expansions, thereby establishing a solid measure-theoretic foundation for deep learning theory.

The weak analogy between Shannon entropy and signed measures, coupled with the lack of geometric characterization for information sets, hinders a structural understanding of information. Method: We propose the Logarithmic Decomposition (LD) framework, which represents the information structure of random variables as definable “logarithmic atoms” over the sample space Ω. By extending Yeung’s I-measure, integrating signed measure theory with information geometry, and introducing the notion of logarithmic decomposability, the framework enables structured criteria for positive and negative entropy atoms. Contribution/Results: LD geometrically reconstructs common information and sufficient statistics; strictly distinguishes dyadic from triadic systems—beyond the capability of I-measure; unifies set-theoretic interpretations of mutual information, conditional entropy, and related quantities; establishes a foundation for quality-oriented information theory; and admits a natural extension to continuous distributions.

Existing conformal prediction (CP) methods provide finite-sample validity guarantees but lack the ability to quantify support strength for arbitrary events and do not support posterior inference. Method: We propose model-free generalized fiducial inference (MF-GFI), the first framework unifying desiderata-based inference with confidence set approximation via optimal probability measures that approximate fuzzy belief/likelihood pairs, with reliability and accountability as core inferential principles. The method strictly controls Type-I error in finite samples and enables both exact and approximate probabilistic reasoning. We develop a computationally tractable probability approximation algorithm yielding prediction sets with rigorous error guarantees. Contribution/Results: This work establishes the first model-free, accountable, and broadly applicable statistical framework for uncertainty quantification in machine learning—offering finite-sample validity, event-specific support assessment, and principled posterior-like inference without distributional assumptions.

Traditional probabilistic laws face persistent challenges—including indeterminate boundaries of physical possibility and acute empirical underdetermination. Method: This paper proposes a novel metaphysical framework grounded in algorithmic randomness, introducing a “probability-constrained law” paradigm that replaces generative chance laws. It jointly employs Kolmogorov complexity and relative frequency as dual constraints, and integrates nonstandard probability models with possible-worlds semantics to rigorously delimit the set of physically possible histories. Contribution/Results: The work achieves the first systematic synthesis of algorithmic information theory with a neo-Humean (i.e., non-Humean) conception of laws, thereby resolving one class of empirical underdetermination while uncovering and characterizing a previously overlooked type. It substantially enhances both the empirical testability and metaphysical constraint strength of probabilistic laws, providing a critical pathway toward a unified account of non-Humean laws.

This paper addresses the categorical decomposition of probabilistic structures in Markov categories, establishing a rigorous categorical foundation for absolute continuity, support sets, and idempotent splittings. Methodologically, it introduces, for the first time, an idempotent splitting theorem for measurable Markov kernels within the category of standard Borel spaces, and distills a general splitting criterion applicable to arbitrary Markov categories. The main contributions are: (1) a precise internal categorical definition of support sets; (2) a proof that every idempotent measurable Markov kernel between standard Borel spaces admits a splitting; and (3) a rigorous, broadly applicable theoretical framework for structural decomposition of probabilistic models, categorical modeling of stochastic processes, and abstract Bayesian inference.

This work addresses the challenge of verifying mathematical proofs generated by large language models by formally encoding, for the first time, an entire advanced undergraduate probability textbook—including its measure-theoretic foundations—into Lean. To bridge the semantic gap between the textbook’s exposition and the abstract formalism of the Mathlib library, the authors introduce an “interface lemma” strategy. Combined with structured proof engineering and formalization techniques specific to measure theory, this approach yields a reusable, machine-verifiable infrastructure spanning fourteen textbook chapters. The resulting formalization not only provides rigorous verification of all stated theorems and explicit articulation of their assumptions but also establishes a robust foundation for reliable AI-assisted mathematics, educational applications, and future formalization efforts in probability theory.

This work addresses why the Kullback–Leibler (KL) divergence is uniquely suited for inference by formalizing inference as the selection of a minimal element within a preorder of positive measures, where divergences serve merely as numerical representations. Building on the axiom of reconstruction invariance—which requires that inference outcomes remain unchanged under equivalent problem formulations—the authors show that KL divergence emerges uniquely without invoking additional assumptions. This framework unifies maximum entropy, Bayesian updating, and exponential family estimation, extending classical axiomatic characterizations from finite alphabets to general measurable spaces. By integrating category theory, f-divergence theory, preorder structures, and Čencov’s category of statistical models, the paper establishes a rigorous mathematical foundation wherein inference operators arise naturally as covariant functors, applicable uniformly across both discrete and continuous settings.

This work addresses the lack of a unified theoretical foundation in traditional uncertainty quantification (UQ), which hinders systematic distinction between epistemic and aleatoric uncertainty. The authors propose a subjective risk decomposition framework that conceptualizes uncertainty as an emergent consequence of modeling choices. For the first time, this framework formally decomposes the two uncertainty types using strictly proper loss functions—such as reversed cross-entropy—grounded in rigorous statistical principles. By integrating information-theoretic analysis with excess risk decomposition from statistical learning theory, the approach not only recovers established uncertainty measures but also establishes novel theoretical connections between uncertainty quantification and the foundations of statistical learning.

When pointwise consistency is unattainable, what is the strongest achievable convergence criterion in hypothesis testing? This work proposes a topological framework of weakened consistency, requiring estimators to converge to the true parameter on a “large” set of probability measures—specifically, a dense set—rather than everywhere. By leveraging topological notions such as comeager sets and dense subsets, the paper establishes a realizability theorem that is strictly weaker than pointwise consistency and demonstrates that, under finite precision, convergence on a dense set necessarily entails inconsistency on a comeager set. This result exposes fundamental limitations inherent in distribution-free hypothesis testing (e.g., conditional independence tests) and offers a novel topological perspective on statistical learnability.

This work addresses the challenge of uncertainty quantification in Poisson signal models with background noise by proposing a confidence interval construction grounded in the principle of Bayesian evidence and the framework of relative belief inference. The method achieves both Bayesian interpretability and frequentist coverage guarantees without requiring prior information, while preserving likelihood ordering consistency and rigorously attaining the prescribed coverage probability. In benchmark scenarios commonly encountered in particle physics, the proposed intervals outperform the widely used Feldman–Cousins approach, thereby offering superior statistical performance. Notably, this is the first method to successfully unify a Bayesian evidential interpretation with strict frequentist coverage properties.