What is the stationary distribution in a Markov chain and does it mean the system stops moving?
The stationary distribution describes the long-run frequency of visiting each state in a Markov chain. It does not mean the system stops moving; rather, it represents a dynamic equilibrium where the probability flows between states balance out. The system continues to transition, but the overall probability mass in each state remains constant.
Conditions
- The Markov chain has reached its long-run behavior.
- The chain is ergodic (irreducible and aperiodic) for a unique stationary distribution to exist.
Reasoning, step by step
- Define the stationary distribution as the long-run frequency of state visits.
- Clarify that the system does not stop moving.
- Explain that it is a dynamic equilibrium where inflows and outflows of probability balance.
- Conclude that the probability mass in each state remains constant over time.
Example
The video shows a bar chart of probabilities for states 0 through 4 and states: 'Not because the system stops moving. But because the flows between states balance out.'
Common misconceptions
- Thinking that a stationary distribution implies the system has stopped changing.
- Believing that the system freezes in one state once stationarity is reached.
Watch the explanation
2:27 – 2:40Watch this moment ↗
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