Wikipedia:Reference desk/Archives/Mathematics/2015 February 9

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February 9

Approximation of hidden markov model by probability distribution

This question is regarding the extent to which a hidden markov model can be approximated by just a single probability distribution of emission probabilities.

An eigenvector of the transition matrix with eigenvalue 1 corresponds to a steady state. Let us assume that the steady state is unique (up to linear multiples). This steady state vector can be normalized into a probability distribution, and the components of this distribution can be used as weights to the emission probabilities, to make the weighted average of the emission probabilities, which gives a single probability probability distribution of the emission probabilities.

Assuming that we have an appropriate distance measure between HMMs, and that the steady state is unique (up to linear multiples), what would be a good statistic, on the transition and emission probability matrices, to determine how approximatable a HMM is, by a degenerate single-state HMM with the distribution of the emission probabilities computed as described above? Thanks! --RM — Preceding unsigned comment added by 203.199.213.3 (talk) 10:16, 9 February 2015 (UTC)

I can think of two trivial cases: First, the emission probabilities are the same for all states, in which case the HMM can be approximated by that emission distribution. Second, the transition matrix is (1/n) times the all-ones matrix, where n is the number of hidden states, in which case the unweighted average of all emission probabilities will give the emission distribution. I'm looking for a generalization of these two cases. --RM — Preceding unsigned comment added by 203.199.213.3 (talk) 10:29, 9 February 2015 (UTC)