Multivariate Dependence and the Sarmanov-Lancaster Expansion

We extend the work of Sarmanov and Lancaster to obtain an expansion for multivariate distributions. The expansion reveals a flexible, detailed dependence structure that goes beyond pairwise linear correlation to include non-linear and higher-order statistical dependencies. Through examples we show how to use the expansion to analyze existing distributions and to construct new distributions with given properties. We also provide a related dependence measure which we decompose into the separate contributions from each subset of random variables. Using the decomposition we analyze neural population data, revealing significant dependencies not captured by cross-correlation.