mathematics

This work formalizes Markov kernels and their foundational applications in probability theory and information theory within the Lean mathematical library Mathlib. Methodologically, it leverages measure theory and dependent type theory to define Markov kernels in the Lean theorem prover, formally verifying their basic properties, disintegration theorem, construction of conditional probabilities and posterior distributions, and unified characterizations of independence, conditional independence, and sub-Gaussian random variables. The contributions are threefold: (i) the first mechanized reconstruction of independence and conditional independence grounded uniformly in Markov kernels; (ii) the first verified formal framework for sub-Gaussian random variables; and (iii) the first rigorous, machine-checked derivation and verification of entropy and the Kullback–Leibler divergence. Collectively, these results establish a sound, coherent, and extensible formal foundation for probabilistic reasoning, Bayesian analysis, and information-theoretic reasoning.

This paper addresses the incompatibility between the random-variable perspective in probability theory and the Markov category framework. To resolve this, we introduce a construction of an abstract sample space category based on Simpson’s axioms, systematically deriving—within any suitable Markov category—a sample space category that is both structurally complete and semantically expressive. Our key contributions are threefold: (i) the first deep integration of Simpson’s probability monad with Markov categories; (ii) a categorical definition of abstract conditional independence; and (iii) the first probabilistic semantic compositional framework taking random variables as primitives. This framework uniformly reconstructs diverse applications—including probabilistic databases and nominal sets—and recovers and generalizes all known Simpson models. It establishes a foundational categorical basis for probabilistic programming, uncertainty reasoning, and structured semantic modeling.

This work addresses the lack of a purely logical characterization and formal semantic foundation for Bayesian inference. We propose a proof-theoretic modeling framework based on multivalued linear logic, establishing—for the first time—a rigorous correspondence between Bayesian networks and multiplicative proof nets: joint probability distributions are semantically encoded as proof structures, and Bayesian inference is reformulated as proof reduction. Leveraging categorical semantics, we uncover a computational isomorphism between probabilistic graphical models and structured proofs. The resulting framework provides a purely logical representation of Bayesian inference, enabling verifiable probabilistic computation. Moreover, it delivers the first formal semantics for probabilistic programming languages grounded in linear logic, thereby bridging the theoretical gap between probabilistic reasoning and logical computation.

This work establishes a universal fluctuation and precision-bound theory for entropy production (EP) in nonequilibrium thermodynamics of stochastic systems with multivariate interactions represented by Bayesian networks. Methodologically, it introduces, for the first time, conditional fluctuation theorems and the thermodynamic uncertainty relation (TUR) into the Bayesian network thermodynamics framework, integrating stochastic processes, information geometry, and large-deviation analysis to derive exact fluctuation theorems for EP of arbitrary subsystem collections—and their conditional EP—and to formulate a novel TUR linking global EP to the precision of probability currents across constituent systems. Key contributions are: (1) quantitative characterization of how directed causal structure constrains thermodynamic precision limits; (2) derivation of a family of universal fluctuation theorems and structure-dependent TURs; and (3) provision of a scalable theoretical foundation for cooperative thermodynamic control in multi-agent systems.

This work addresses the tractability of exact inference and learning in exponential-family latent variable models (LVMs), seeking to characterize the precise boundary of models admitting closed-form analytical solutions without approximation. Method: We derive necessary and sufficient conditions for prior–posterior conjugacy in exponential-family LVMs, providing the first systematic characterization of exact solvability. We further propose a composable graphical model construction framework that preserves structural flexibility while guaranteeing analytic tractability throughout. A general-purpose exact Bayesian inference and parameter learning algorithm is developed, accompanied by an open-source implementation supporting empirical validation across diverse models. Contribution/Results: Our results substantially broaden the class of LVMs amenable to exact inference—bypassing variational approximations or Monte Carlo sampling—and establish a rigorous theoretical foundation and practical toolkit for interpretable, high-precision latent-variable modeling.

This work proposes a novel paradigm that unifies the entire statistical inference pipeline through a probabilistic language, aiming to coherently bridge observed data, inferential targets, and real-world decision-making. By treating probability and stochastic processes as a central “translation language,” the framework integrates tools from probability measures, likelihood theory, weak convergence, empirical processes, functional data analysis, M- and Z-estimation, kernel methods, and event-time processes into a common syntax. This synthesis connects classical theoretical foundations with modern data structures and practical applications. The approach is validated through historical and biomedical case studies, demonstrating its capacity to provide systematic modeling pathways for complex data while substantially enhancing inferential stability and predictive performance.

This work proposes the first fast and universal random number generation algorithm that covers the entire parameter space of the Pearson Type IV distribution, which has long lacked an efficient sampling method due to its complex parameter structure. By employing a carefully designed transformation combined with an adaptive rejection sampling strategy, the algorithm achieves uniformly efficient sampling across all admissible shape parameters. The method not only fills a critical gap in existing computational techniques for this distribution but also demonstrates practical utility in Bayesian inference tasks, confirming its effectiveness in real-world statistical modeling. As such, it provides a key computational tool for implementing complex probabilistic models involving the Pearson Type IV distribution.

This work addresses the lack of explicit confidence modeling for various sources of uncertainty in Bayesian inference by proposing a general extension framework that, for the first time, explicitly incorporates confidence in key uncertainty components—such as the prior and likelihood—into Bayesian modeling. The framework not only introduces a novel regularization mechanism but also provides a unified approach to inducing model sparsity. Without compromising theoretical rigor, the method achieves controllable sparsity across diverse models, including linear regression, logistic regression, and Bayesian neural networks, thereby significantly enhancing both interpretability and generalization performance.