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provides the probabilities for observing events given a particular state. In the above example, event 1 will be observed 90% of the time if we are in state 1 while event 2 has a 10% probability of occurring in this state. In contrast, event 1 will only be observed 20% of the time if we are in state 2 and event 2 has an 80% chance of occurring. Given an arbitrary row-vector describing the state of the system (<math>\mathbf{\pi}</math>), the probability of observing event j is then:
:<math>\mathbf{P}(O = j)=\sum_{i} \pi_i b_{
This can be represented in matrix form by multiplying the state row-vector (<math>\mathbf{\pi}</math>) by an observation matrix (<math>\mathbf{O_j} = \mathrm{diag}(b_{*,o_j})</math>) containing only diagonal entries. Each entry is the probability of the observed event given each state. Continuing the above example, an observation of event 1 would be:
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