A General Theory of Multivariable Dependence: Statistical Connectivity, Path Allocations, and Structural Computation
Year: 2026
Type: preprint
Abstract: This work develops a measure-theoretic, inferential, and computational framework for multivariable statistical dependence. For a finite collection of random elements, statistical connectivity is defined as the relative entropy of the joint probability law from the product of its coordinate marginals. Regular conditional probability kernels extend this definition to background-dependent connectivity fields while preserving the distinction between pointwise conditional quantities and their integrated values. The finite-set theory establishes projective consistency, nonnegativity, coordinate invariance, data processing, independent-block additivity, and exact decompositions along paths in Boolean lattices. Averaging marginal increments over uniformly sampled permutation paths yields the Shapley allocation. More general path laws and coalition distributions produce probability-weighted allocations. Möbius inversion, low-order interaction models, sparse supports, maximum-entropy projections, and latent subspace representations provide complementary descriptions of higher-order dependence and structured cooperative games. The statistical theory covers finite tables, continuous variables, complex samples, missing observations, measurement channels, stochastic processes, nonregular boundaries, and high-dimensional selection. Population functionals are systematically separated from estimators, numerical approximations, and uncertainty procedures. Structured algorithms combine coalition-mass truncation, interaction truncation, low-rank approximation, path sampling, state aggregation, and transport-based compression with explicit error bounds. Topological, homological, spectral, sheaf-theoretic, and fibered constructions are introduced only after their coefficient systems, filtrations, and transport maps have been specified. The framework maintains strict boundaries among statistical dependence, Möbius interaction, discrete square circulation, geometric curvature, transport holonomy, spectral differentials, estimation error, and causal effects. Without additional structural assumptions, unrestricted black-box games and dependence systems retain exponential information and computational complexity. Keywords multivariable dependence; statistical connectivity; total correlation; relative entropy; conditional probability kernels; Boolean lattices; path decomposition; Shapley allocation; Möbius inversion; higher-order interaction; cooperative games; probability-weighted coalitions; structural sparsity; low-rank approximation; Monte Carlo inference; transport geometry; spectral sequences; statistical computation.
Author: Wang, Kianming
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7203720738
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