Hidden markov models HMM

What is a Markov Model?

A stochastic model describing a sequence of possible events where the probability of each event depends only on the state of the previous event.

What is the Markov property?

The future state depends only on the current state, not on the sequence of events that preceded it.

What is an Observable Markov Model?

A model where each state is directly associated with an observable symbol.

What is a Hidden Markov Model (HMM)?

A statistical model where the system being modeled is assumed to be a Markov process with hidden states.

What are the components of an HMM?

A set of states, observation symbols, transition probabilities, emission probabilities, and initial state probabilities.

What is the transition matrix in an HMM?

A matrix describing the probabilities of transitioning from one state to another.

What is the emission matrix in an HMM?

A matrix describing the probabilities of observing each symbol from each state.

What is the initial state distribution in an HMM?

A vector describing the probabilities of starting in each state.

What is the joint probability of a sequence in an HMM?

The probability of a sequence of observations and the corresponding sequence of hidden states.

What is the likelihood of an observation sequence in an HMM?

The sum of the joint probabilities over all possible state sequences.

What is the Forward Algorithm?

An algorithm to efficiently compute the likelihood of an observation sequence.

What is the time complexity of the Forward Algorithm?

O(N^2 * T), where N is the number of states and T is the length of the sequence.

What is the Backward Algorithm?

An algorithm to compute the probability of the remaining observations given a current state.

What is the time complexity of the Backward Algorithm?

O(N^2 * T).

What is the Viterbi Algorithm?

An algorithm to find the most probable sequence of hidden states given an observation sequence.

What is the time complexity of the Viterbi Algorithm?

O(N^2 * T).

What is the Baum-Welch Algorithm?

An algorithm used to estimate the parameters of an HMM using the Expectation-Maximization (EM) approach.

What is the time complexity of the Baum-Welch Algorithm?

O(N^2 * T * I), where I is the number of iterations.

What is the Expectation-Maximization (EM) Algorithm?

A general algorithm for maximum likelihood estimation with hidden variables.

What is Jensen’s Inequality in the EM Algorithm?

An inequality used to show that maximizing the expected log-likelihood improves the likelihood.

What is the E-step in the EM Algorithm?

Computing the expected value of the complete data log-likelihood.

What is the M-step in the EM Algorithm?

Maximizing the expected log-likelihood with respect to the parameters.

What is the difference between a generative model and a discriminative model?

A generative model models the joint distribution, while a discriminative model models the conditional distribution.

What is a bigram model?

A Markov model where the probability of a state depends only on the previous state.

What is an n-gram model?

A Markov model where the probability of a state depends on the previous n-1 states.

What is the difference between a first-order and a second-order Markov model?

A first-order model depends on the previous state, while a second-order model depends on the previous two states.

What is the assumption of stationarity in a Markov model?

The transition probabilities do not depend on time.

What is a stochastic process?

A collection of random variables representing a process evolving over time.

What is a sequence likelihood in an HMM?

The probability of observing a sequence of symbols given the model.

What is the state sequence in an HMM?

A sequence of hidden states corresponding to an observation sequence.

What is the best path in an HMM?

The state sequence with the highest probability given an observation sequence.

What is the posterior probability of a state in an HMM?

The probability of being in a particular state at a given time, given the observation sequence.

What is the probability of a transition in an HMM?

The probability of moving from one state to another at a given time.

What is the observation probability in an HMM?

The probability of observing a particular symbol from a state.

What is sequence classification using HMMs?

Assigning a class to a sequence by choosing the HMM with the highest likelihood.

What is a positive training set in HMM learning?

A set of sequences that represent examples from the target concept.

What is the log-likelihood in HMMs?

The natural logarithm of the likelihood, used for numerical stability.

Why is log-likelihood used instead of likelihood?

To prevent numerical underflow when dealing with small probabilities.

What is a stochastic finite state automaton?

A finite state machine where transitions are governed by probabilities.

What is an observable sequence in HMMs?

A sequence of symbols that can be directly observed.

What is a hidden state sequence in HMMs?

A sequence of states that is not observed but generates the observable sequence.

What is model convergence in the Baum-Welch Algorithm?

The point at which the model parameters stabilize.

What is the initialization problem in HMM training?

The sensitivity of the Baum-Welch Algorithm to initial parameter values.

What is a likelihood plateau in HMM training?

A situation where the likelihood stops increasing significantly.

What is the forward probability in HMMs?

The probability of reaching a state given the observations up to that time.

What is the backward probability in HMMs?

The probability of observing the remaining sequence from a state.

What is the difference between the Forward and Viterbi Algorithms?

The Forward Algorithm computes the likelihood, while the Viterbi Algorithm finds the most probable path.

What is a partial observation sequence in HMMs?

An observation sequence that is not complete.

What is path probability in the Viterbi Algorithm?

The probability of the most likely path up to a given state.

What is a transition probability matrix?

A matrix where each entry represents the probability of transitioning between states.