Overlapping Gaussian output distributions for HMM states The emission probabilities of a 2-state HMM model have overlapping Gaussian distributions with equal mean values. If the observed data sequence X is given, is it possible to infer the state sequence of the Markov process which generated the observed data?

 A: A Gaussian is not only represented by its mean but also by its variance. The variances will be very different for each Gaussian in this case and they are taking into account in the HMM. Basically, if each of these Gaussian is associated with one hidden state of the HMM, the likelihood of a data sample with respect to each of the state will be different.
Even out of an HMM context, you can intuitively see that a sample $X=5$ has "more chances" of having been generated by the red Gaussian than by the blue one. More formally its likelihood with respect to the red Gaussian is greater that its likelihood with respect to the blue Gaussian.
$P(X=5|\mu_{red}, \sigma_{red}) > P(X=5|\mu_{blue}, \sigma_{blue})$
Opposite to this a sample $X=0$ will have a higher likelihood with respect to the blue Gaussian.
These likelihood are taken into account when training an HMM as well as when using the Viterbi algorithm, that allows for the estimation of the most probable sequence of hidden states given an observation sequence.
