Variable-order Markov model: Difference between revisions

Revision as of 22:43, 20 April 2007

Revision as of 22:48, 20 April 2007

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Consider a sequence with the [[Markov chain|Markov property]] <math>x_1^{n}=x_1x_2...x_n</math> of <math>n</math> realizations of [[random variable]]s, where <math> x_i\in A</math> is the state (symbol) at position <math>i</math> 1≤<math>i</math>≤<math>n</math>, and the concatenation of states <math>x_i</math> and <math>x_{i+1}</math> is denoted by <math>x_ix_{i+1}</math>.

Given a training set of observed states, <math>x_1^{n}</math>, the construction algorithm of the VOM models [2,3,4] learns a model <math>P</math> 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 distribution|conditional probability distribution]] <math>P(xx_i|s)</math> for a symbol <math>x_i \in A</math> given a context <math>s\in A^*</math>, where the * sign represents a sequence of states of any length, including the empty context.

VOM models attempt to estimate [[conditional distribution]]s of the form <math>P(xx_i|s)</math> where the context length |<math>s</math>|≤<math>D</math> varies depending on the available statistics.

In contrast, conventional [[Markov chain|Markov models]] attempt to estimate these [[conditional distribution]]s by assuming a fixed contexts' length |<math>s</math>|=<math>D</math> and, hence, can be considered as special cases of the VOM models.