Transition matrix

Definition

A transition matrix is a square matrix used to describe the probabilities of moving from one state to another in a Markov chain. Each entry in the matrix represents the probability of transitioning from a particular state to another state, and the sum of each row equals one. This structure allows for the analysis of state changes and helps understand the long-term behavior of the system being studied.

5 Must Know Facts For Your Next Test

1. The entries in a transition matrix must be non-negative and each row must sum up to one, reflecting the total probability for each state transition.
2. The transition matrix can be used to compute future state probabilities by multiplying it with the current state distribution vector.
3. The concept of transition matrices applies not only to discrete-time Markov chains but also to continuous-time Markov processes.
4. By analyzing the transition matrix, one can derive important properties such as irreducibility and periodicity of the Markov chain.
5. In the context of absorption and ergodicity, transition matrices help identify which states can absorb others and determine long-term probabilities for various states.

Review Questions

• How do transition matrices facilitate the understanding of Markov chains and their properties?
  • Transition matrices are essential for analyzing Markov chains because they provide a structured way to represent the probabilities of transitioning between different states. Each entry in the matrix corresponds to a specific probability, which allows researchers to visualize and compute future state distributions based on current conditions. This visualization aids in identifying key properties such as irreducibility and periodicity, which are crucial for understanding how the Markov chain behaves over time.
• Discuss how you would use a transition matrix to find long-term behavior in a Markov chain.
  • To find the long-term behavior in a Markov chain using a transition matrix, you would typically look for a stationary distribution. This involves raising the transition matrix to a large power or using iterative methods to see how the probabilities converge as time progresses. When done correctly, this process reveals the stable probabilities associated with each state in the Markov chain, indicating how likely it is for the system to be found in each state after many transitions.
• Evaluate the role of transition matrices in determining absorbing states and their implications for ergodicity in Markov chains.
  • Transition matrices play a crucial role in identifying absorbing states by showing which states lead into others without leaving. In a matrix where some rows have all entries zero except for one entry that equals one, we can identify an absorbing state. Understanding these properties is significant when examining ergodicity; if every state can eventually lead to an absorbing state, it implies that over time, certain long-term behaviors will stabilize, allowing predictions about final outcomes based on initial conditions.

Related terms

Markov chain: A stochastic process that undergoes transitions from one state to another within a finite or countably infinite set of possible states, characterized by the property that the future state depends only on the current state.

Stationary distribution: A probability distribution over states that remains unchanged as time progresses, meaning that once the system reaches this distribution, it will continue to follow it indefinitely.

Absorbing state: A state in a Markov chain that, once entered, cannot be left; the probability of remaining in that state is one.