Markov Bases and Beyond

In many applications (hypothesis testing and disclosure limitation) one wants to investigate the set of probability measures or contingency tables satisfying a given set of linear constraints. The main examples for such constraints are fixed expectation values or marginal distributions and conditional distributions of subsystems. Markov bases can be used to explore these sets. However, the use of Markov bases may not always be applicable, for example when the linear constraints are not given by integral equations. Some ideas how to generalize the Markov bases technique are presented.