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Computation Complexity and Forward Algorithm in Hidden Markov Models

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Computation Complexity and Forward Algorithm in Hidden Markov Models

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Questions and Answers

What is the goal of the likelihood computation in hidden Markov models?

To find the optimal state sequence that maximizes the likelihood of the observation sequence

To estimate the parameters of the hidden Markov model

To determine the probability of the observation sequence given the hidden Markov model (correct)

To determine the most likely state sequence given the observation sequence

What is the formula used to compute the likelihood of the observation sequence $X = (x_1, x_2, ..., x_T)$ given the hidden Markov model $M$?

$p(X|M) = \prod_{t=1}^T P(x_t|q_t)$

$p(X|M) = P(q_1)P(x_1|q_1) \prod_{t=2}^T P(q_t|q_{t-1})P(x_t|q_t)$

$p(X|M) = \sum_{Q\in Q} P(X, Q|M)$ (correct)

$p(X|M) = \max_{Q\in Q} P(X, Q|M)

How many possible state sequences $Q = (q_1, q_2, ..., q_T)$ are there for an observation sequence of length $T$ in a hidden Markov model with $N$ states?

$N$

$N^2$

$T^N$

$N^T$ (correct)

Which of the following is a key challenge in computing the likelihood of the observation sequence in a hidden Markov model?

The need to sum over all possible state sequences Signup and view all the answers

What is the purpose of the forward algorithm in hidden Markov models?

To compute the likelihood of the observation sequence given the hidden Markov model Signup and view all the answers

What is the purpose of the Viterbi algorithm in hidden Markov models?

To find the most likely state sequence given the observation sequence Signup and view all the answers

Which of the following is a key difference between the forward algorithm and the Viterbi algorithm in hidden Markov models?

The forward algorithm computes the likelihood, while the Viterbi algorithm finds the most likely state sequence Signup and view all the answers

What is the role of the observation probability $P(x_t|q_t)$ in the computation of the likelihood of the observation sequence in a hidden Markov model?

It represents the probability of observing $x_t$ given that the current state is $q_t$ Signup and view all the answers

What is the role of the transition probability $P(q_t|q_{t-1})$ in the computation of the likelihood of the observation sequence in a hidden Markov model?

It represents the probability of transitioning from state $q_{t-1}$ to state $q_t$ Signup and view all the answers

What is the purpose of the initial state probability $P(q_1)$ in the computation of the likelihood of the observation sequence in a hidden Markov model?

It represents the probability of the initial state $q_1 Signup and view all the answers