On symmetric partition lattices and probability interactions

Abstract

The Möbius function of a partition lattice can be used to define interactions of probability measures. We call a lower set a symmetric partition lattice when it is invariant under permutations of the variables. We characterize these lattices through integer partitions and obtain a recursion for their Möbius coefficients at the top element. By constructing probability measures whose partition products are linearly independent, we show that permutation invariance of an interaction is equivalent to that of its underlying lower set, provided the coordinate spaces are sufficiently large. We also characterize the partitions on a lower set that force an interaction to vanish for every probability measure, and construct indecomposable distributions with zero interaction. We define the order of a symmetric partition lattice and relate it to vanishing marginals. We study generalized Lancaster and define the Streitberg and size-limited partition lattices, obtaining explicit Möbius coefficients for them. Finally, we express interaction equations in terms of characteristic functions. For Gaussian distributions, we show that an interaction vanishes exactly when the distribution factorizes according to a nontrivial partition in the underlying lower set. We also give a nonvanishing criterion for radial characteristic functions generated by Bernstein functions.