POS Tagging - Viterbi Decoding
Introduction to Viterbi Decoding for POS Tagging
Part-of-Speech (POS) tagging is a fundamental task in Natural Language Processing that involves assigning grammatical categories (noun, verb, adjective, etc.) to words in a sentence. The Viterbi algorithm is a dynamic programming approach that efficiently solves the decoding problem in Hidden Markov Models (HMMs) to find the most probable sequence of POS tags for a given sentence.
Hidden Markov Models for POS Tagging
In the context of POS tagging, an HMM consists of:
- Hidden States: POS tags (noun, verb, adjective, etc.)
- Observable Symbols: Words in the sentence
- Transition Probabilities: P(tag_i | tag_i-1) - likelihood of transitioning from one POS tag to another
- Emission Probabilities: P(word | tag) - likelihood of a word being generated by a specific POS tag
The Viterbi Algorithm
The Viterbi algorithm solves the decoding problem: given a sequence of words and the HMM parameters (transition and emission probabilities), find the most likely sequence of POS tags that generated those words.
Mathematical Foundation
For a sentence with words w₁, w₂, ..., wₙ and possible tags t₁, t₂, ..., tₘ, we want to find:
argmax P(tag_sequence | word_sequence)
Using Bayes' theorem and the Markov assumption, this becomes:
argmax ∏ P(word_i | tag_i) × P(tag_i | tag_i-1)
Dynamic Programming Approach
The Viterbi algorithm uses a table (Viterbi matrix) where:
- Rows: Represent possible POS tags
- Columns: Represent words in the sentence
- Cells: Store the maximum probability of any tag sequence ending at that tag for that word
Algorithm Steps
Initialization: For the first word, compute:
V[tag][1] = π[tag] × emission[tag][word₁]where π[tag] is the initial probability of the tag.
Recursion: For each subsequent word position j and each tag:
V[tag][j] = max(V[prev_tag][j-1] × transition[prev_tag][tag]) × emission[tag][word_j]Backtracking: Trace back through the stored path pointers to recover the optimal tag sequence.
Example Walkthrough
Consider the sentence "Book a park" with tags {Noun, Verb, Det}:
Step 1: Emission Matrix (P(word|tag))
Book a park
Noun 0.3 0.1 0.7
Verb 0.5 0.0 0.1
Det 0.0 0.9 0.0
Step 2: Transition Matrix (P(tag_j|tag_i))
Noun Verb Det
Noun 0.2 0.1 0.7
Verb 0.6 0.1 0.3
Det 0.8 0.2 0.0
Step 3: Viterbi Table Computation
For "Book":
- V[Noun][1] = 0.3 (emission only, assuming uniform start)
- V[Verb][1] = 0.5
- V[Det][1] = 0.0
For "a":
- V[Noun][2] = max(0.3×0.2, 0.5×0.6, 0.0×0.8) × 0.1 = 0.03
- V[Verb][2] = max(0.3×0.1, 0.5×0.1, 0.0×0.2) × 0.0 = 0.0
- V[Det][2] = max(0.3×0.7, 0.5×0.3, 0.0×0.0) × 0.9 = 0.189
And so on for "park"...
Computational Complexity
- Time Complexity: O(N × T²) where N is sentence length and T is number of tags
- Space Complexity: O(N × T) for the Viterbi table
Without dynamic programming, finding the optimal path would require O(T^N) time, making the Viterbi algorithm essential for practical applications.
Applications Beyond POS Tagging
The Viterbi algorithm is widely used in:
- Speech Recognition: Finding most likely word sequences from acoustic signals
- Bioinformatics: Gene sequence analysis and protein structure prediction
- Information Extraction: Named entity recognition and sequence labeling
- Machine Translation: Alignment of word sequences between languages
Key Insights
- Optimal Substructure: The optimal solution contains optimal solutions to subproblems
- Markov Property: Future states depend only on the current state, not the entire history
- Greedy Choice: At each step, we choose the locally optimal solution that leads to the global optimum
- Memory Efficiency: Only previous column values are needed for computation
The Viterbi algorithm elegantly balances local word-tag compatibility (emission probabilities) with global sequence coherence (transition probabilities) to find the most linguistically plausible POS tag sequence.