Definition of Markov Bases and Other Bases

In this chapter we define Markov bases and introduce various basic concepts and properties of Markov bases. In Chapter 1 we have already discussed the notion of Markov bases for two-way and three-way contingency tables. In this chapter Markov bases are defined for general discrete exponential family models. We also clarify the relations between moves and binomials and state the fundamental fact that a Markov basis is a set of generators of a toric ideal. Besides Markov basis, we introduce two other bases, a lattice basis and the Graver basis. There are some overlaps of this chapter with the previous chapter, since the previous chapter was intended as a reference.