Cumulant varieties

For a discrete distribution in Rd on a finite support D probabilities and moments are algebraically related. If there are n=D support points then there are n probabilities p(x),x D and n basic moments. By suitable interpolation of the probabilities using a Gröbner basis method, high order moments can be express linearly in terms of n basic moments. A main result is that high order cumulants can also be expressed as polynomial functions of n low order moments and cumulants. This means that statistical models which can be expressed via an algebraically variety for the basic probabilities and moments, such as graphical models, induce a variety for the basic cumulants, which we shall call the “cumulant variety”. It is important to stress that the cumulant variety depends on the monomial ordering defining the original Gröbner basis.