Hilbert space embeddings of independence tests of several variables with radial basis functions

In this paper we characterize the continuous radial basis functions that can be used to determine if a probability measure is the product of its marginals, that is, the class of radial basis functions that can be used as independence tests. We also characterize the continuous radial basis functions that can be used to determine whether the Lancaster/Streitberg interaction of a probability measure is zero. To fill a gap between these two probabilistic contexts, we introduce an indexed measure of independence that generalizes the Lancaster interaction. We also characterize the continuous radial basis functions that can be used to determine whether this new indexed Lancaster interaction is zero. In all cases, these functions are extensions of the class of Bernstein functions to several variables. We present examples of these functions using high-order completely monotone functions.