Hidden Markov Models

Definition

Hidden Markov Models (HMMs) are statistical models that represent systems which follow a Markov process with hidden states. They are characterized by the assumption that the system being modeled is a Markov process, but the states themselves are not directly observable, making them 'hidden.' HMMs are widely used in various fields such as speech recognition, bioinformatics, and finance to analyze time series data where the underlying state transitions cannot be directly measured.

5 Must Know Facts For Your Next Test

1. HMMs consist of hidden states and observable outputs, where transitions between hidden states are governed by probabilities.
2. Each hidden state in an HMM can produce different observable outputs according to specific emission probabilities.
3. The initial state distribution in an HMM indicates the probability of starting in each hidden state before any observations are made.
4. HMMs can be trained using algorithms like the Baum-Welch algorithm, which adjusts model parameters based on observed data to maximize the likelihood of the observed sequence.
5. Applications of HMMs include natural language processing for part-of-speech tagging and bioinformatics for gene prediction.

Review Questions

• How do Hidden Markov Models differ from traditional Markov Chains in terms of observability and state representation?
  • Hidden Markov Models differ from traditional Markov Chains primarily in that HMMs involve hidden states that cannot be observed directly. In a standard Markov Chain, all states and transitions are visible and can be analyzed straightforwardly. However, in HMMs, only the outputs or emissions corresponding to the hidden states are observable, requiring additional methods to infer the underlying state sequence based on these observations.
• Discuss how emission probabilities function within Hidden Markov Models and their importance in determining observable outputs.
  • Emission probabilities in Hidden Markov Models are crucial as they determine how likely it is for a given hidden state to produce certain observable outputs. Each hidden state has its own set of emission probabilities, which connects the invisible structure of the model with actual observations. This relationship allows for understanding how certain states might influence or generate specific events in observed data, making these probabilities essential for tasks such as decoding and prediction.
• Evaluate the role of the Viterbi algorithm in Hidden Markov Models and how it contributes to inferring hidden states from observable data.
  • The Viterbi algorithm plays a pivotal role in Hidden Markov Models by providing an efficient method for determining the most likely sequence of hidden states that results in a given sequence of observable outputs. It utilizes dynamic programming to compute probabilities systematically, thus avoiding redundant calculations. This approach not only speeds up the inference process but also enhances accuracy in applications such as speech recognition and biological sequence analysis, where understanding the underlying state transitions is key to interpreting the observed data.

Related terms

Markov Chain: A Markov chain is a mathematical system that undergoes transitions from one state to another on a state space, where the probability of each transition depends only on the current state.

Emission Probability: In an HMM, emission probabilities define the likelihood of observing a particular output from a given hidden state, linking the hidden states with observable events.

Viterbi Algorithm: The Viterbi algorithm is an efficient dynamic programming algorithm used to find the most probable sequence of hidden states in a Hidden Markov Model given a sequence of observed events.