Markov Chains are mathematical models used to predict a sequence of events where future states depend solely on the present state. They are characterized by state spaces and transition probabilities, encapsulated in a transition matrix. These models find applications in finance, computer science, genetics, and meteorology. Key properties like aperiodicity, irreducibility, and ergodicity define their behavior, while the Markov Chain Monte Carlo method leverages them in complex probability sampling.
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Markov Chains are stochastic processes where the probability of each event depends only on the state attained in the preceding event
Transition Matrix
The transition matrix is a key tool in analyzing the behavior of Markov Chains, representing the probabilities of transitioning from one state to another
Deriving Transition Probabilities
Transition probabilities can be determined from empirical data or theoretical models, such as a weather prediction model
Steady-State Distributions
Steady-state distributions, which are stable over time, can be identified by examining the transition matrix
Absorbing States
Absorbing states, from which the system cannot exit once entered, can also be identified through the transition matrix
Markov Chains are used in finance to model and predict price movements
Markov Chains are used in computer science to predict user navigation patterns, such as the Google PageRank algorithm
Markov Chains are used in genetics to model the evolution of gene sequences over time
Markov Chains are used in meteorology for weather prediction
Aperiodic Markov Chains do not follow a fixed cycle of states, allowing for a more diverse range of transitions
Irreducible Markov Chains permit transitions between any two states, reflecting a connected system
Ergodic Markov Chains, which are both aperiodic and irreducible, converge to a stable long-term distribution
Absorbing Markov Chains include at least one state that, once entered, cannot be left, modeling processes with a certain conclusion
The Markov Chain Monte Carlo method is a computational algorithm that uses Markov Chains and Monte Carlo simulation techniques to sample from complex probability distributions
MCMC is particularly useful in Bayesian statistics for estimating characteristics of challenging distributions
MCMC is used in various fields, including statistical inference, financial modeling, artificial intelligence, and climate science
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The ______ ______ is a crucial instrument for examining Markov Chains, representing the odds of moving from one state to another.
transition matrix
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Constructing Transition Matrix
Enumerate system states, determine transition probabilities from data or models.
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Steady-State Distribution
Long-term probability distribution over states, remains constant over time.
