Example
Consider for example a sequence of random variables, each of which takes a value from the ternary alphabet {a, b, c}. Specifically, consider the string aaabcaaabcaaabcaaabc...aaabc constructed from infinite concatenations of the sub-string aaabc.
The VOM model of maximal order 2 can approximate the above string using only the following five conditional probability components: {Pr(a|aa) = 0.5, Pr(b|aa) = 0.5, Pr(c|b) = 1.0, Pr(a|c)= 1.0, Pr(a|ca)= 1.0}.
In this example, Pr(c|ab) = Pr(c|b) = 1.0; therefore, the shorter context b is sufficient to determine the next character. Similarly, the VOM model of maximal order 3 can generate the string exactly using only five conditional probability components, which are all equal to 1.0.
To construct the Markov chain of order 1 for the next character in that string, one must estimate the following 9 conditional probability components: {Pr(a|a), Pr(a|b), Pr(a|c), Pr(b|a), Pr(b|b), Pr(b|c), Pr(c|a), Pr(c|b), Pr(c|c)}. To construct the Markov chain of order 2 for the next character, one must estimate 27 conditional probability components: {Pr(a|aa), Pr(a|ab), ..., Pr(c|cc)}. And to construct the Markov chain of order three for the next character one must estimate the following 81 conditional probability components: {Pr(a|aaa), Pr(a|aab), ..., Pr(c|ccc)}.
In practical settings there is seldom sufficient data to accurately estimate the exponentially increasing number of conditional probability components as the order of the Markov chain increases.
The variable-order Markov model assumes that in realistic settings, there are certain realizations of states (represented by contexts) in which some past states are independent from the future states; accordingly, "a great reduction in the number of model parameters can be achieved."
Read more about this topic: Variable-order Markov Model
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