I recently started learning HMM and was wondering how do I go about using a model or similar thereof in which an observation is really a realization of the Gaussian distribution of the corresponding hidden states? (Each hidden state would correspond to a Gaussian distribution with unique mean, standard deviation,...)

I suppose the state transition probability doesn't have to be continuous.

Are there any models based on this? or similar to this?

  • $\begingroup$ Do you mean that you would like to generate data from an HMM? (which is totally possible) I am not sure I get your question... And yes, you are right, usually state transitions are "controlled" by a probability mass distribution but some extended models propose to modify these probabilities as well (for instance nonstationary HMMs) $\endgroup$ – Eskapp Dec 8 '16 at 15:59
  • $\begingroup$ I guess my "model" would be that there are multiple gaussian distributions and each hidden state would correspond to a unique linear combination of them with coefficients representing probability of observing, aka emitting, that corresponding gaussian distribution. Speaking of non-stationary HMM, by the way, how would I go about learning that? I can't seem to find relevant notes about them $\endgroup$ – Kevvy Kim Dec 9 '16 at 15:56
  • $\begingroup$ About non-stationary HMMs, they are also called Variable Duration HMMs. Here are two references: (1) P. M. Djuric, J.-H. Chun, An MCMC sampling approach to estimation of nonstationary hidden Markov models, IEEE Trans. on Signal Proc. 50(5), 2002, 1113–1123. and (2) L. R. Rabiner, A tutorial on hidden Markov models and selected applications in speech recognition, Proceedings of the IEEE 77(2), 1989, 257–286. Are you trying to generate some data from an HMM you would design yourself or are you trying to train an HMM from some data that you have? $\endgroup$ – Eskapp Dec 15 '16 at 20:28
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    $\begingroup$ I found what I was looking for. It's gaussian mixture HMM. $\endgroup$ – Kevvy Kim Dec 20 '16 at 8:57

I don't think that HMM is anything to do with distribution of hidden probabilities. Though distribution of response variable matters. Transition probabilities could come through forward backward algo. if there are no co-variates then fixed matrix would come else it would be functions of PDF for all states( as a function of external variables/co-variates).

  • $\begingroup$ to clarify, I guess I'm looking for a model in which each hidden state would correspond to a unique linear combo of various gaussian PDFs with coefficients representing the probability of observing that gaussian. $\endgroup$ – Kevvy Kim Dec 9 '16 at 15:58
  • $\begingroup$ If there are any external variables( co-variates) with the visible states. transition probabilities would be a function of external variable/s also. What do you exactly mean by model here? $\endgroup$ – Arpit Sisodia Dec 10 '16 at 12:52
  • $\begingroup$ I found what I was looking for. It's gaussian mixture HMM. $\endgroup$ – Kevvy Kim Dec 20 '16 at 8:57

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