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I'm applying the EM to a hidden markov chain (the $\mathbf{Z}=\{Z_1,...,Z_n\}$ variable), with observations(the $\mathbf{Y}=\{Y_0,...,Y_n\}$ variable) dependent not only on the hidden markov chain, but also on the immediate past observation (the $Y_{j-1}$).

$$Log(P(\mathbf{Z},\mathbf{Y}|\mathbf{\Theta)})=\log(\pi_{0z_1})+\sum^n_{j=2}\log(\pi_{z_{j-1}z_j})+\sum^n_{j=1}\log(P(y_j|y_{j-1},z_j;\mathbf{\Theta)})$$

In "McLachlan's Finite Mixtures", page 63 and 64, it talks about how to obtain the Observed Information Matrix, in terms of the complete data Log-Likelihood. My question is how, from the Observed Information Matrix, do we get the standard errors? (Why can one estimate the inverse of the covariance matrix of the MLE, by the Observed Information Matrix?) And could I obtain the std errors in a similar way for a hidden Markov model?

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    $\begingroup$ The Observed information matrix under certain "regularity conditions" can be a consistent estimator of the Fisher Information matrix. I tried to prove this in a post here $\endgroup$ – dandar Jul 24 '15 at 16:42
  • $\begingroup$ I am not sure if this theory carries straight over to the complete data observed information matrix too. I imagine it does since the EM algorithm by maximising the complete data LL also maximises the ordinary LL. However I would want to see some theory on this and I am no expert. $\endgroup$ – dandar Jul 24 '15 at 16:48

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