## Podcast Beta

## Questions and Answers

Question 1

What is the time complexity of the search space for possible state sequences in an HMM using the Viterbi algorithm? Provide the formula.

Answer 1

O(N^T), where N is the number of states and T is the length of the observation sequence.

Question 2

What is the purpose of the trellis in the Viterbi algorithm?

Answer 2

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Question 3

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## Study Notes

Viterbi Algorithm for Hidden Markov Models: Finding the Best Hidden State Sequence

- The search space for possible state sequences X is O(N^T) given an observation sequence O and an HMM μ = (A, B).
- The Viterbi algorithm is used to find the most likely hidden state sequence in an HMM given a sequence of observations. It employs dynamic programming and reduces computational complexity by storing and reusing the results of previously solved subproblems.
- The Viterbi algorithm calculates the probability of reaching each state once for each time step and memoizes it in a trellis, which is a dynamic programming table. The trellis has states as rows and time steps as columns, and each cell records the Viterbi probability δj(t) - the probability of the most likely path that ends in state s_j at time t.
- The main step of the Viterbi algorithm uses a helper variable ψj(t) to store the t - 1 state index i on the highest probability path. In the backtracing phase, ψ is used to find the previous cell/state in the best path.
- The Viterbi algorithm begins with an initialization step where the probability of a state starting the sequence at t = 0 is just the probability of it emitting the first symbol.
- The intuition behind the Viterbi algorithm is that it only needs to record one previous step for a first-order HMM because the HMM has a limited horizon. This reduces the computational effort to O(N^2 T) for a first-order HMM.
- The trellis is the main data structure used in the Viterbi algorithm. It has states as rows and time steps as columns, with each cell (j, t) recording the Viterbi probability δj(t).
- The Viterbi algorithm is used to determine when each type of dice was used in the dishonest casino dice example. The two hidden states are L for loaded dice and F for fair dice.
- The optimal substructure property and the overlapping subsolutions property enable the Viterbi algorithm to efficiently compute the most likely state sequence that explains the observations given the HMM parameters.
- In HMMs, it is necessary to keep N states at each time step in order to ensure that we can find the most likely state sequence overall and avoid getting stuck with a suboptimal sequence due to a greedy choice.
- Reducing the table to just a one-dimensional table of T time slots would be a greedy choice and might not produce the optimal solution.
- By keeping all N states at each time step, we maintain a record of the most likely paths for each state up to that point. This allows us to consider all possible state sequences and find the most likely state sequence overall.

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## Description

Test your knowledge of the Viterbi Algorithm for Hidden Markov Models with this quiz! Learn about the properties of the algorithm, its main data structure, and how it is used to find the most likely hidden state sequence given a sequence of observations. Discover how the algorithm can efficiently compute the solution and avoid getting stuck with a suboptimal sequence due to a greedy choice. Challenge yourself with questions on the initialization step, trellis structure, and backtracing phase. Take the quiz now and