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I have a learned Hidden Markov Model (HMM) from a certain sequential data using Gibbs sampling. I have managed to obtain the transition probabilities (transition matrix) of the Markov chain and the params of the probability distributions of the hidden states. The only set of params that I have yet been able to obtain are the initial probabilities $\pi$ that specify where the Markov chain would start. So given the parameters that I have estimated, is there a way to figure out $\pi$? Or how can I specify the Gibbs sampler to find out $\pi$?

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    $\begingroup$ Baum-Welch is capable of estimating the initial probabilities along with the transition and emission matrices. But I guess you opted not to use it. Is there a specific reason why the initial probabilities are important? $\endgroup$
    – Zhubarb
    Dec 18, 2013 at 8:38
  • $\begingroup$ I understand that this can be done easily with EM algorithm. The reason why I'm using Gibbs sampling here is because I want to incorporate covariates into $\pi$ and the transition matrix $T$, which I know how to specify using the BUGS modeling language. I don't know how to do this using EM. $\endgroup$
    – Joe
    Dec 18, 2013 at 9:17

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If you're training the HMM on one long string then it only has one example of a transition from the start state, thus your initial transition probability is rather meaningless. To get a meaningful estimate of $\pi$, you must break the sequence into natural segments and precede each segment with a distinguished 'start state', then train the model.

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