Some problems of categorical data and stochastic optimization involve
studying probability distributions on nonnegative integer lattice points
on multidimensional hyperplanes. Simple formulas in such situations
are scarce. However, simple Markov chains can be run on such sets which
allow one in to analyze any distribution. The general construction
is the work of Diaconis and Sturmfels and uses Groebner bases. We
will discuss some concrete examples and problems.