Definition
Let be a state space (finite alphabet) of size |A|.
Consider a sequence with the Markov property of realizations of random variables, where is the state (symbol) at position 1≤≤, 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 that leads to a better variance-bias tradeoff of the learned models.
Read more about this topic: Variable-order Markov Model
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