Markov Models and State Transitions
Markov Models are mathematical frameworks used to model systems that transition between states over time, where the probability of moving to the next state depends only on the current state, not on the sequence of events that preceded it. State transitions refer to the movement from one state to another within this model. Markov Models are widely used in fields like statistics, computer science, and economics to predict sequences and analyze random processes.
Markov Models and State Transitions
Markov Models are mathematical frameworks used to model systems that transition between states over time, where the probability of moving to the next state depends only on the current state, not on the sequence of events that preceded it. State transitions refer to the movement from one state to another within this model. Markov Models are widely used in fields like statistics, computer science, and economics to predict sequences and analyze random processes.
💡 Key Takeaways
- Understand the memoryless property: the future depends only on the current state, not on the past trajectory.
- Learn how to model a system with states and a transition probability matrix, and read or build a simple Markov chain from data.
- Differentiate between discrete-time Markov chains and continuous-time Markov processes and know when each is appropriate.
- Explore long-run behavior: stationary (steady-state) distributions, convergence to equilibrium, and the role of absorbing states in modeling termination.
❓ Frequently Asked Questions
What is a Markov model?
A mathematical framework for systems that move between states over time with transition probabilities that depend only on the current state, not on past history.
What is the Markov property?
The future state depends only on the present state, not on previous states or how the process arrived there.
What is a transition matrix?
A matrix where entry P_ij is the probability of moving from state i to state j in one step; each row sums to 1.
What is a stationary distribution?
A probability distribution over states that remains unchanged after transitions: pi = pi P. It exists (and is unique) for certain Markov chains.
What is an absorbing state?
A state that, once entered, cannot be left (P_ii = 1 and P_ij = 0 for j ≠ i).