What is the difference between Forward-backward algorithm and Viterbi algorithm?

Forward-Backward Algorithm vs. Viterbi Algorithm: A Comprehensive Comparison

Both the Forward-Backward and Viterbi algorithms are essential tools in the realm of Hidden Markov Models (HMMs). While they share a common foundation in HMMs, they differ significantly in their objectives and applications. This article aims to illuminate the distinctions between these two algorithms, providing a comprehensive understanding of their functionalities and applications.

Understanding Hidden Markov Models (HMMs)

Before delving into the algorithms, it’s crucial to grasp the core concept of Hidden Markov Models. An HMM describes a system with:

• Hidden States: Unobservable states of the system, denoted as ‘S’.
• Observations: Observable outputs generated by the system, denoted as ‘O’.
• Transition Probabilities: Probabilities of transitioning between hidden states. This is represented by the matrix ‘A’.
• Emission Probabilities: Probabilities of emitting observations from each hidden state. This is represented by the matrix ‘B’.

Forward-Backward Algorithm

Objective:

The Forward-Backward algorithm, also known as the Baum-Welch algorithm, focuses on calculating the probability of a sequence of observations, given a specific HMM model. It aims to estimate the parameters (transition and emission probabilities) of the model that best explain the observed data.

Procedure:

• Forward Pass: Calculate the probability of observing the first ‘t’ observations, ending in state ‘i’. This is denoted as α(t,i).
• Backward Pass: Calculate the probability of observing the remaining observations, starting from state ‘i’ at time ‘t’. This is denoted as β(t,i).
• Parameter Estimation: Update the model parameters (A and B) using the computed α and β values to maximize the likelihood of the observed sequence.

Application:

The Forward-Backward algorithm is primarily used for:

• Model Training: Estimating the model parameters given a set of training data.
• Model Evaluation: Assessing the performance of a trained model on unseen data.

Viterbi Algorithm

Objective:

The Viterbi algorithm, unlike the Forward-Backward algorithm, aims to determine the most likely sequence of hidden states that generated a given sequence of observations. It seeks to find the best path through the hidden state space that maximizes the overall probability.

Procedure:

• Initialization: Assign a probability to each hidden state at time ‘t=1’.
• Recursion: For each time step ‘t’, calculate the probability of reaching state ‘i’ from all previous states. Select the path with the highest probability.
• Termination: Find the state with the highest probability at the final time step. Trace back through the selected paths to reconstruct the most likely hidden state sequence.

Application:

The Viterbi algorithm is widely used in:

• Speech Recognition: Decoding the most likely sequence of phonemes from an audio signal.
• Sequence Alignment: Identifying the best alignment between two sequences, such as DNA or protein sequences.
• Machine Translation: Finding the most probable translation of a sentence in another language.

Comparison Table

| Feature | Forward-Backward Algorithm | Viterbi Algorithm |
|—|—|—|
| Objective | Calculate the probability of a sequence of observations | Find the most likely hidden state sequence |
| Output | Probability of observations | Most likely hidden state sequence |
| Algorithm | Forward and backward passes | Dynamic programming |
| Application | Model training and evaluation | Decoding and prediction |

Example Code (Python):

Forward-Backward Algorithm:

import numpy as np

# Define HMM parameters
A = np.array([[0.7, 0.3], [0.4, 0.6]])  # Transition probabilities
B = np.array([[0.8, 0.2], [0.2, 0.8]])  # Emission probabilities
pi = np.array([0.5, 0.5])  # Initial state probabilities

# Define observation sequence
observations = [0, 1, 0]

# Calculate alpha and beta values (Forward and Backward passes)
alpha = np.zeros((len(observations), 2))
beta = np.zeros((len(observations), 2))

# Forward pass
alpha[0, :] = pi * B[:, observations[0]]

for t in range(1, len(observations)):
    for i in range(2):
        alpha[t, i] = np.sum(alpha[t - 1, :] * A[:, i] * B[i, observations[t]])

# Backward pass
beta[-1, :] = 1

for t in range(len(observations) - 2, -1, -1):
    for i in range(2):
        beta[t, i] = np.sum(beta[t + 1, :] * A[i, :] * B[:, observations[t + 1]])

# Calculate probability of observations
probability = np.sum(alpha[-1, :] * beta[-1, :])
print("Probability of observations:", probability)

Viterbi Algorithm:

import numpy as np

# Define HMM parameters
A = np.array([[0.7, 0.3], [0.4, 0.6]])  # Transition probabilities
B = np.array([[0.8, 0.2], [0.2, 0.8]])  # Emission probabilities
pi = np.array([0.5, 0.5])  # Initial state probabilities

# Define observation sequence
observations = [0, 1, 0]

# Initialize viterbi matrix and backpointer matrix
viterbi = np.zeros((len(observations), 2))
backpointer = np.zeros((len(observations), 2), dtype=int)

# Initialization step
viterbi[0, :] = pi * B[:, observations[0]]
backpointer[0, :] = -1  # No previous state

# Recursion step
for t in range(1, len(observations)):
    for i in range(2):
        viterbi[t, i] = np.max(viterbi[t - 1, :] * A[:, i] * B[i, observations[t]])
        backpointer[t, i] = np.argmax(viterbi[t - 1, :] * A[:, i] * B[i, observations[t]])

# Termination step
best_state = np.argmax(viterbi[-1, :])

# Traceback to reconstruct path
hidden_states = [best_state]
for t in range(len(observations) - 2, -1, -1):
    best_state = backpointer[t + 1, best_state]
    hidden_states.insert(0, best_state)

print("Most likely hidden state sequence:", hidden_states)

In conclusion, the Forward-Backward and Viterbi algorithms are distinct yet complementary techniques within the framework of HMMs. The former focuses on model training and evaluation, while the latter excels in decoding and predicting hidden states. Choosing the appropriate algorithm depends on the specific task and desired outcome.