Viterbi Algorithm

We can calculate the optimal path in a hidden Markov model using a dynamic programming algorithm. This algorithm is widely known as Viterbi Algorithm. Viterbi [10] devised this algorithm for the decoding problem, even though its more general description was originally given by Bellman [3].

Let X be a sequence of length L. for $k \in Q$ and $0 \leq i \leq L$, we consider a path $\Pi$ ending at k, and the probability of $\Pi$ generating the prefix $(x_{1},\ldots,x_{i})$ of X. Denote by v_k(i) the maximal (over all paths, $\Pi$, ending at k) such probability.

$$v_{k}(i) = \max_{\{\Pi\vert\Pi_{i}=k\}}{ P(x_{1},\ldots,x_{i} \vert \Pi)}$$ (11)

1.

Initialize:

v_begin(0) = 1 (12)

$$\forall_{k \neq begin} \quad v_{k}(0)$$ = 0 (13)

2.

For each $i=0,\ldots,L-1$ and for each $l \in Q$ recursively calculate:

$$v_{l}(i+1) = e_{l}(x_{i+1}) \cdot \max_{k \in Q} {\{v_{k}(i) \cdot a_{kl}\}}$$ (14)

3.

Finally, the value of $P(X\vert\Pi^{*})$ is:

$$P(X\vert\Pi^{*}) = \max_{k \in Q}{\{v_{k}(L) \cdot a_{k,end}\}}$$ (15)

We can reconstruct the path $\Pi^{*}$ itself by keeping back pointers during the recursive stage and tracing them.

Complexity: We calculate the values of $O(\vert Q\vert \cdot L)$cells of the matrix V, spending O(|Q|) operations per cell. The overall time complexity is therefore $O(L \cdot \vert Q\vert^2)$ and the space complexity is $O(L \cdot \vert Q\vert)$.

Since we are dealing with probabilities, the extensive multiplication operations we perform may result in an underflow. This can be avoided if we choose to work with logarithmic scores. We can therefore define v_k(i) to be the logarithmic score of the most probable path for the prefix $(x_{1},\ldots,x_{i})$ that ends in the state k.

We shall initialize:

v_begin(0) = 0 (16)

$$\forall_{k \neq begin} \quad v_{k}(0) -\infty$$ = (17)

The recursion will look like:

$$v_{l}(i+1) = \log{e_{l}(x_{i+1})} + \max_{k \in Q} {\{v_{k}(i) + \log(a_{kl})\}}$$ (18)

Finally, the score for the best path $\Pi^{\ast}$ is:

$$Score(X\vert\Pi^{*}) = \max_{k \in Q}{\{v_{k}(L) + \log(a_{k,end})\}}$$ (19)