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So a hidden markov model consists of hidden states $H_i \in \{1,2,...,n\}, i \in \{1,...,\infty\}$, observable states $O_j \in \{1,...,p\}, j \in \{1,...,\infty \}$, transition probabilities $P(X_i=x|X_{i-1}=y), \forall i\in\{2,...,\infty\}, \forall x,y \in \{1,...,n\}$, emission probabilities $P(O_j=k|H_j=m), \forall k\in\{1,...,p\}, \forall m \in \{1,...,n\} $.

The probability of observing a sequence $x_1,...,x_k$ is $P(x_1,...,x_k)=\sum_{i_1}P(O_1=x_1|H_1=i_1)P(H_1=i_1)\sum_{i_2}P(O_2=x_2|H_2=i_2)P(H_2=i_2|H_1=i_1)...\sum_{i_k}P(O_k=x_k|H_k=i_k)P(H_k=i_k|H_{k-1}=i_{k-1})$

Fist question is is the calculation of probability correct?

Second question is can you show this computation as matrix products(I know it can)?

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