We begin in column 1 (for the word "Janet") by setting the Viterbi value in each cell to the product of the $$\pi$$ transition probability and the observation likelihood of the word "Janet" given the tag for that cell. Most of the cells in the column are zero since the word "Janet" cannot be any of those tags. Next, each cell in the "will" column gets updated. For each state, we compute the value $$viterbi[s,t]$$ by taking the maximum over the extensions of all the paths from the previous column that lead to the current cell according to: $$v_t(j) = \max_{N \leq i \leq n} v_{t-1}(i) a_{ij} b_j(o_t)$$ The remaining value is multiplied by the relevant observation probability, and the (trivial) max is taken.

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This table lists the ai j probabilities for transitioning between the hidden states (part-of-speech tags)

example

The Viterbi algorithm first sets up a probability matrix, with one column for each observation $o_t$ and one row for each state in the state graph. Each column thus has a cell for each state $q_i$ in the single combined automaton. Each cell of the matrix is given by:
$$v_t(j) = \max_{q_1, ..., q_{t-1}} P(q_1...q{t-1}, o_1, ..., o_t, q_t = j|\lambda)$$

For a given state $q_j$ at time $t$, we compute the value $v_t(j)$ as:
$$v_t(j) = \max_{i=1}^N v_{t-1}(i)a_{ij}b_j(o_t)$$
where:
- $v_{t-1}(i)$ is the previous Viterbi path probability from the previous time step
- $a_{ij}$ is transition probability from previous state $q_i$ to current state $q_j$
- $b_j(o_t)$ is the state observation likelihood of the observation symbol $o_t$ given the current state $j$

Details of the Viterbi Algorithm

Viterbi Algorithm Matrix Computation Example

The viterbi model can be bidirectional.
For example, the tagger can be run twice, once left-to-right and once right-to-left. In Viterbi decoding, the labeler would choose the higher scoring of the two sequences.


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