Hidden Markov Models

Definition

A Hidden Markov Model (HMM) is a statistical model used to represent systems that are assumed to follow a Markov process with unobserved (hidden) states. HMMs are used to model the probabilistic relationships between sequences of observable events and the hidden states that generate them.

Key Concepts

States: The hidden variables in the system. Each state represents a possible situation or condition of the system.
Observations: The visible outputs of the system, which are probabilistically related to the hidden states.
Transition Probabilities: The probabilities of transitioning from one state to another.
Emission Probabilities: The probabilities of observing a particular output given a state.
Initial State Probabilities: The probabilities of the system starting in each state.
Forward Algorithm: A recursive algorithm used for calculating the probability of a sequence of observations.
Backward Algorithm: A recursive algorithm used for calculating the probabilities of the initial parts of the state sequence given future observations.
Viterbi Algorithm: An algorithm for finding the most probable sequence of hidden states given a sequence of observations.
Baum-Welch Algorithm: An algorithm for estimating the parameters of an HMM.

Detailed Explanation

Components of an HMM:
- States (( S )): A set of hidden states ( {S_1, S_2, ..., S_N} ).
- Observations (( O )): A set of possible observations ( {O_1, O_2, ..., O_M} ).
- Transition Matrix (( A )): A matrix ( A ) where ( A[i, j] ) is the probability of transitioning from state ( S_i ) to state ( S_j ).
- Emission Matrix (( B )): A matrix ( B ) where ( B[j, k] ) is the probability of observing ( O_k ) given state ( S_j ).
- Initial State Distribution (( \pi )): A vector ( \pi ) where ( \pi[i] ) is the probability of starting in state ( S_i ).
Algorithms for Inference and Learning:
- Forward Algorithm: Calculates the probability of a sequence of observations by summing over all possible state sequences.
  - Initialization: ( \alpha_1(i) = \pi[i] \cdot B[i, O_1] )
  - Recursion: ( \alpha_{t+1}(j) = \left( \sum_{i=1}^{N} \alpha_t(i) \cdot A[i, j] \right) \cdot B[j, O_{t+1}] )
  - Termination: ( P(O|\lambda) = \sum_{i=1}^{N} \alpha_T(i) )
- Backward Algorithm: Calculates the probability of the ending part of the sequence given the states.
  - Initialization: ( \beta_T(i) = 1 )
  - Recursion: ( \beta_t(i) = \sum_{j=1}^{N} A[i, j] \cdot B[j, O_{t+1}] \cdot \beta_{t+1}(j) )
  - Termination: ( P(O|\lambda) = \sum_{i=1}^{N} \pi[i] \cdot B[i, O_1] \cdot \beta_1(i) )
- Viterbi Algorithm: Finds the most probable sequence of hidden states.
  - Initialization: ( \delta_1(i) = \pi[i] \cdot B[i, O_1] )
  - Recursion: ( \delta_{t+1}(j) = \max_{i} \left( \delta_t(i) \cdot A[i, j] \right) \cdot B[j, O_{t+1}] )
  - Termination: ( P^* = \max_{i} \delta_T(i) )
- Baum-Welch Algorithm: An iterative algorithm for parameter estimation.
  - Expectation Step: Compute the expected values of the hidden variables.
  - Maximization Step: Maximize the likelihood of the observed data given the expected values.
Example:
- Scenario: Weather prediction
- States: ( Sunny, Rainy )
- Observations: ( Dry, Wet )
- Transition Matrix: [ A = \begin{bmatrix} 0.8 & 0.2 \ 0.3 & 0.7 \end{bmatrix} ]
- Emission Matrix: [ B = \begin{bmatrix} 0.9 & 0.1 \ 0.3 & 0.7 \end{bmatrix} ]
- Initial State Distribution: [ \pi = \begin{bmatrix} 0.6 & 0.4 \end{bmatrix} ]
- Inference Tasks:
  - Use the forward algorithm to calculate ( P(O) ) for a given sequence of observations ( O ).
  - Use the Viterbi algorithm to find the most likely sequence of hidden states given the observations.

Diagrams

Example of a Hidden Markov Model

Transition and Emission Matrices

States	Sunny	Rainy
Sunny	0.8	0.2
Rainy	0.3	0.7

Observations	Dry	Wet
Sunny	0.9	0.1
Rainy	0.3	0.7

Links to Resources

Notes and Annotations

Summary of key points: Hidden Markov Models are powerful tools for modeling time-series data where the system's states are hidden. Key algorithms for HMMs include the forward and backward algorithms for filtering and smoothing, the Viterbi algorithm for decoding, and the Baum-Welch algorithm for learning.
Personal annotations and insights: Mastery of HMMs is crucial for applications such as speech recognition, bioinformatics, and financial modeling. Understanding these models provides a solid foundation for more complex probabilistic models and time-series analysis techniques.

Backlinks

Artificial Neural Networks: HMMs can be combined with neural networks for hybrid models that handle sequential data.
Data Science: Time-series analysis and anomaly detection benefit from the probabilistic reasoning capabilities of HMMs.
Natural Language Processing: HMMs are foundational for tasks like part-of-speech tagging and speech recognition.