8.2 Markov chains: definition, transition probabilities, and steady-state distributions

Markov chains are stochastic processes where the future state depends only on the current state. They're key to modeling systems with memoryless properties, from weather patterns to customer behavior.

Transition probabilities between states are represented in matrices. By analyzing these, we can predict long-term behavior and find steady-state distributions, crucial for understanding system dynamics over time.

Markov chains and their properties

Definition and classification

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• Markov chains are stochastic processes satisfying the Markov property, which states that the future state depends only on the current state, not on past states (memoryless property)
• Classified as discrete-time or continuous-time Markov chains based on whether state transitions occur at fixed time intervals (discrete-time) or random times (continuous-time)
• State space of a Markov chain can be finite (e.g., number of customers in a queue) or infinite (e.g., stock prices), and states can be discrete or continuous
• Time-homogeneous Markov chains have transition probabilities that remain constant over time

Essential properties

• Memoryless property enables modeling complex systems without considering the entire history of the process
• Complete characterization of a Markov chain requires an initial state distribution and a transition probability matrix
• Irreducibility and aperiodicity are important properties for determining the existence and uniqueness of a steady-state distribution
  • Irreducible Markov chains allow reaching any state from any other state in a finite number of steps
  • Aperiodic Markov chains have a greatest common divisor of return times to any state equal to 1

Transition probability matrices

Definition and properties

• Transition probability matrix P is a square matrix containing probabilities of transitioning from one state to another in a single step
• Element Pij represents the probability of moving from state i to state j in one step
• Rows of a transition probability matrix sum to 1, representing probabilities of transitioning from a given state to all possible states
• Markov chain is completely characterized by its initial state distribution and transition probability matrix

Examples

• In a weather model, states could be "sunny," "cloudy," or "rainy," with transition probabilities based on historical data
• For a simple two-state Markov chain (e.g., "healthy" and "sick"), the transition probability matrix would be a 2x2 matrix with probabilities of staying in the same state or transitioning to the other state

Chapman-Kolmogorov equations

Calculating n-step transition probabilities

• Chapman-Kolmogorov equations calculate the probability of transitioning from one state to another in n steps
• n-step transition probability matrix P^n is obtained by multiplying the one-step transition probability matrix P by itself n times
• The (i,j)th element of P^n represents the probability of moving from state i to state j in exactly n steps

Long-term behavior and applications

• Chapman-Kolmogorov equations derive the long-term behavior of a Markov chain, such as the steady-state distribution
• Useful for predicting the long-run proportion of time spent in each state (e.g., market share in a brand loyalty model)
• Can be used to calculate the mean first passage time, which is the expected number of steps to reach a specific state for the first time

Steady-state distribution of Markov chains

Definition and existence conditions

• Steady-state distribution (also called stationary or equilibrium distribution) is a probability distribution that remains unchanged under the transition matrix
• Markov chain has a steady-state distribution if, as the number of steps tends to infinity, the probability of being in each state converges to a fixed value, regardless of the initial state distribution
• Steady-state distribution π satisfies the equation π = πP, where P is the transition probability matrix
• Markov chain has a unique steady-state distribution if it is irreducible and aperiodic

Calculation methods

• Solve the system of linear equations π = πP, subject to the constraint that the elements of π sum to 1
• Eigenvalue method: The steady-state distribution is the left eigenvector of the transition probability matrix corresponding to the eigenvalue 1
• Power method: Repeatedly multiply an initial probability distribution by the transition probability matrix until convergence

Applications of Markov chains

Population dynamics and ecology

• Model the evolution of a population over time, with states representing age groups (e.g., juveniles, adults, seniors) or health conditions (e.g., susceptible, infected, recovered)
• Analyze the long-term behavior of ecosystems, with states representing ecological conditions (e.g., high, medium, low biodiversity) or species abundances

Marketing and customer behavior

• Model customer behavior, such as brand loyalty or switching patterns, with states representing different brands (e.g., Coca-Cola, Pepsi) or products
• Predict long-term market shares and identify effective marketing strategies based on the steady-state distribution

Finance and economics

• Model transitions between different economic states, such as bull and bear markets, or credit ratings of bonds (e.g., AAA, AA, A, BBB)
• Assess the long-term risk and stability of financial assets or economic systems

Natural language processing

• Model the probability of a word or sequence of words occurring based on the previous words in the sequence (e.g., bigram or trigram models)
• Generate text or assess the likelihood of a given sentence based on the learned transition probabilities between words or phrases