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Backward Algorithm (Hidden Markov Model)

About the Algorythm Recipe 🍰

Imagine you're trying to guess what the last few days' weather was like based on people's reactions today, like if they're carrying an umbrella. The backward algorithm in a hidden Markov model helps by calculating the probability of being in a particular state (like sunny, rainy, or cloudy) at a specific time in the past, given the future observations. It's like working backward from today's clues to piece together what might have happened before!

Cookin' time! 🍳


import numpy as np

def backward_algorithm(emission_probs, initial_probs, transition_probs):
  """
  This function computes the backward probabilities for a given HMM.

  Args:
      emission_probs: A numpy array of shape (num_states, num_emissions) representing the emission probabilities.
      initial_probs: A numpy array of shape (num_states,) representing the initial state probabilities.
      transition_probs: A numpy array of shape (num_states, num_states) representing the transition probabilities.

  Returns:
      A numpy array of shape (num_states, sequence_length) representing the backward probabilities for each state in the sequence.
  """

  # Get the number of states and sequence length
  num_states = emission_probs.shape[0]
  sequence_length = len(emission_sequence)  # Assuming emission_sequence is defined elsewhere

  # Initialize the backward probabilities array
  backward_probs = np.zeros((num_states, sequence_length))

  # Set the final backward probabilities (all 1s for the last time step)
  backward_probs[:, -1] = 1

  # Calculate backward probabilities for the remaining time steps (in reverse order)
  for t in range(sequence_length - 2, -1, -1):
    for i in range(num_states):
      # Sum over all next possible states
      backward_probs[i, t] = np.sum(backward_probs[:, t+1] * transition_probs[i, :] * emission_probs[:, emission_sequence[t+1]])

  return backward_probs

# Example usage (replace with your actual data)
# Same data from the Forward Algorithm example can be used here

backward_probs = backward_algorithm(emission_probs, initial_probs, transition_probs)

print(backward_probs)

This code defines a function backward_algorithm that takes the same arguments as the Forward Algorithm. It calculates the backward probabilities for each state in the sequence using a loop that iterates backward through the sequence.


Note:

  • This code snippet only implements the Backward Algorithm. It's often used in conjunction with the Forward Algorithm (e.g., for the Baum-Welch algorithm).

  • This is a basic example, and real-world implementations might involve additional functionalities and optimizations.

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