Forward Algorithm

Finding probability
Definition
Example
Summary

Section 1 - Page 7
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2b. Calculating a's at time t = 1
We calculate partial probabilities as :

a t ( j )= Pr( observation | hidden state is j ) x Pr(all paths to state j at time t)

In the special case where t = 1, there are no paths to the state. The probability of being in a state at t = 1 is therefore the initial probability, i.e. Pr( state | t = 1) = P(state), and we therefore calculate partial probabilities at t = 1 as this probability multiplied by the associated observation probability;

formula
Thus the probability of being in state j at intialisation is dependent on that state's probability together with the probability of observing what we see at that time.