What is Stochastic Supervenience?

Standard formulations of supervenience typically treat higher level properties as point valued facts strictly fixed by underlying base states. However, in many scientific domains, from statistical mechanics to machine learning, basal structures more naturally determine families of probability measures than single outcomes. This paper develops a general framework for stochastic supervenience, in which the dependence of higher level structures on a physical base is represented by Markov kernels that map base states to distributions over macro level configurations. I formulate axioms that secure law like fixation, nondegeneracy, and directional asymmetry, and show that classical deterministic supervenience appears as a limiting Dirac case within the resulting topological space of dependence relations. To connect these metaphysical claims with empirical practice, the framework incorporates information theoretic diagnostics, including normalized mutual information, divergence based spectra, and measures of tail sensitivity. These indices are used to distinguish genuine structural stochasticity from merely epistemic uncertainty, to articulate degrees of distributional multiple realization, and to identify macro level organizations that are salient for intervention. The overall project offers a conservative extension of physicalist dependence that accommodates pervasive structured uncertainty in the special sciences without abandoning the priority of the base level.