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Given a random variable $X$ which arise from a parameterized distribution $F(X;θ)$, the likelihood is defined as proportional to the probability of observed data as a function of $θ$: $\operatorname{L}(θ | x)=\operatorname{P}(X=x \mid θ)$

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Learning hidden Markov model where transition/emission/initial probabilities aren't independent

by just computing the likelihood using the forward algorithm. … However, this has proved slow (my sequence is extremely long) and somewhat unreliable (the likelihood surface is a bit weird looking). …
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