Hidden Markov Models Lecture Notes

Overview of Hidden Markov Models

• Different from previous algorithms (e.g., k-means clustering, linear regression).
• Instead of relying solely on large data sets, hidden Markov models utilize probability distributions.

Weather Prediction Example

• The purpose is to predict the weather based on probabilities of different events occurring.
• Example probabilities:
  • If it's sunny, 80% chance it will be sunny again, 20% chance of rain.
  • Additional information may include average temperature and conditions for sunny and cold days.
• Hidden Markov models can be constructed with predefined probability distributions.

Definition of Hidden Markov Model (HMM)

• HMM consists of:
  • A finite set of states (e.g., hot day, cold day).
  • Each state associated with a multi-dimensional probability distribution.
  • Transitions governed by transition probabilities.

Key Components of HMM

States

• Define the states without direct observation; termed "hidden".
• For example, states like hot and cold days.

Observations

• Each state has outcomes or observations based on probability distributions.
• Example: For hot weather, probability of Tim being happy (80%) or sad (20%).

Transition Probabilities

• Likelihood of moving from one state to another.
  • Hot day to cold day (20%) and hot day to hot day (80%).
  • Cold day to hot day (30%) and cold day to cold day (70%).

Example Illustration

1. Draw two states: hot day (yellow) and cold day (gray).
2. Define transition probabilities between these states.
3. Determine observation distributions for each state:
   • Hot Day: Average temperature of 20°C (between 15°C and 25°C).
   • Cold Day: Average temperature of 5°C (between -5°C and 15°C).

Purpose of HMM

• To predict future events based on past events.
• Use the model to forecast weather over a week based on current weather conditions.
• If today is warm, determine the likelihood of tomorrow being cold.

Conclusion

• Understanding states, transitions, and observations is crucial.
• Predictions rely on the established model using defined probabilities.