In a recent paper, I proposed to study complex systems through a mereological lens by applying the Möbius inversion theorem. I also covered this in a recent blogpost. Here, I will collect some of the most important applications of Möbius transformations in the sciences. I will update this table as I find more applications. If you have suggestions, please send me an email, or leave a comment below!
Field of Study Macro quantity Mereology Micro quantity Statistics Moments Powerset Central moments Moments Partitions Cumulants Free moments Non-crossing partitions Free cumulants Path signature moments Ordered partitions Path signature cumulants Causal effects Antichains Causal synergy/redundancy Information Theory Entropy Powerset Mutual information Entropy Singletons Total correlation Surprisal Powerset Pointwise mutual information Joint Surprisal Powerset Conditional interactions Mutual Information Antichains Synergy/redundancy atom Biology Pheno- & Genotype Powerset Epistasis Gene expression profile Powerset Genetic interactions Population statistics Powerset Synergistic treatment effect Physics Energy Powerset Ising interactions Correlation functions Partitions Ursell functions Quantum corr. functions Partitions Scattering amplitudes Chemistry Molecular property Subgraphs Fragment contributions Molecular property Reaction poset Cluster contributions Game Theory Coalition value Powerset Harsanyi dividends Shapley value Supersets Normalised coalition synergy Artificial Intelligence Generative model probabilities Powerset Feature interaction Predictive model predictions Powerset Feature contribution Dempster-Shafer Belief Lattices Evidence weight KL-divergence Powerset $\Delta_{p|q}$ measure