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So, initially when I was using CTW the problem was with VQ, in particular, independence of the quantization from the inference in the Markovian model. I solved that using HMMs so that the quantization is learned simultaneously with the structure.

Can we use left-to-right HMMs (probably with emissions from a mixture of Gaussians), and still use the number of states as a complexity measure? Or perhaps a combination of \(N_{states}\) and \(N_{mixtures}\)? This might function as an indirect way of representing temporal structure.

I just realized that I can meaningfully compare many HMMs if I sort the states. If I put in initial emission distribution means so that for a univariate case the first state is the one with the lowest mean whereas the last one is the one with the highest mean, then I can ensure that I get the same state labeling for all trained models. This naturally extends to the multivariate case.

Original post is here. I have changed the order of \(\beta_n\) coefficients.

Original post is here.