We begin in column 1 (for the word Janet) by setting the Viterbi value in each cell to the product of the π 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 $vt(j) =\max_{N \leq i \leq n}v_{t−1}(i) a_{ij}b_{j}(o_{t})$
 The remaining value is multiplied by the relevant observation probability, and the (trivial) max is taken.

further explanation

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

University of Michigan - Ann Arbor

The Viterbi algorithm is a probabilistic extension of minimum edit distance. It computes the maximum probability alignment of two strings instead of minimum edit distance.

Viterbi

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

This mathematical expression defines a function, denoted as $$F$$, whose behavior is determined by a set of parameters, specifically $$\omega$$ and $$\hat{\theta}$$. The subscript parameters modify the function's output. The hat symbol (^) over $$\theta$$ typically indicates that it is an estimated parameter, likely derived from data. The notation $$(\cdot)$$ serves as a placeholder for the function's input argument.

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