Suppose you are modeling text with a HMM, What is the complexity of finding most the probable sequence of tags or states from a sequence of text using brute force algorithm?

Posted on February 16, 2019March 7, 2019 by MLNerds

Assume there are total $K$ states and let $T$ be the length of the largest sequence.
Think how we generate text using an hMM. We first have a state sequence and from each state we emit an output. From each state, any word out of $T$ possible outcomes can be generated. Since there are $K$ states, at each possible state in the sequence, there are $T$ choices of generating a word. Hence, there are $T^{K}$ possible sequences
Therefore, complexity using any brute force algorithm is $O(T^{K})$ as we would explore probability of each sequence to find the maximum for most probable one. This is clearly not scalable.
In practice, this problem is solved using the viterbi algorithm using dynamic programming.