Decoding problem of Hidden Markov Model

Decoding problem of Hidden Markov Model, One of the three fundamental problems to be solved under HMM is Decoding problem, Decoding problem is the way to figure out the best hidden state sequence using HMM

Decoding problem of Hidden Markov Model

Decoding problem: 

Given an HMM λ = (A, B, π) and an observation sequence O = o₁, o₂, …, o_T, how do we choose the corresponding optimal hidden state sequence (most likely sequence) Q = q₁, q₂, …, q_T that can best explain the observations.

Explain decoding problem of HMM with example

Decoding problem is the one which is used to uncover the hidden part of the model. It is used to find the best possible hidden state sequence for a given observation sequence. 

Given an observation sequence and a HMM, the task of the decoder is to find the best hidden state sequence.

This can be achieved as discussed in Evaluation problem. That is, compute P(O|Q) for each possible hidden sequence and choose the one with highest probability score as the best state sequence. But the major problem is the number of possible state sequences for a given observation sequence, ie., N^T, where N is number of hidden states and T is number of observations.

• If number of tags N = 2, and number of words observed T = 2, then 2² = 4 possible likelihood estimates.

• If N = 6 and T = 4, then 6⁴ = 1296 possible likelihood estimates.

• For longer sentences, it will be too high.

What is the solution to handle decoding problem of HMM?

• Dynamic programming using the decoding algorithm (Viterbi algorithm).

The Viterbi algorithm solves the decoding problem efficiently by using dynamic programming. Instead of examining all possible state sequences, it builds the optimal path step-by-step by storing the best partial sequence up to each time step. This reduces complexity from exponential to O(N²T).

Example of Decoding in POS Tagging

Consider a simple POS tagging example where an HMM predicts the most likely sequence of tags for words like "eat food". The decoding task determines whether the sequence is more likely to be Verb → Noun or Noun → Verb based on transition and emission probabilities.

Key Takeaways

• Decoding finds the most likely hidden state sequence.
• Direct computation is impossible due to Nᵀ combinations.
• Viterbi algorithm provides an optimal and efficient solution.
• Used widely in POS tagging, speech recognition, and sequence modeling.

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