Let <math>D(s)A</math> be a state space (finite alphabet) of size |DA|. Consider a sequence with the Markov property <math>x<sup>n</sup>=x<sub>1</sub>x<sub>2</sub>...x<sub>n</sub></math> of <math>n</math> realizations of random variables, where is the state (symbol) at position i 1in, and the concatenation of states and is denoted by . Given a training set of observed states, , the construction algorithm of the VOM models learns a model that provides a probability assignment for each state in the sequence given its past (previously observed symbols) or future states. Specifically, the learner generates a conditional probability distribution for a symbol given a context , where the * sign represents a sequence of states of any length, including the empty context. VOM models attempt to estimate conditional distributions of the form where the context length varies depending on the available statistics. In contrast, conventional Markov models attempt to estimate these conditional distributions by assuming a fixed contexts' length and, hence, can be considered as special cases of the VOM models. Effectively, for a given training sequence, the VOM models are found to obtain better model parameterization than the fixed-order Markov Models (Ben-Gal et al, 2005) that leads to a better variance-bias tradeoff of the learned models.
