A Markov chain is a mathematical model that represents a sequence of events or states where the probability of transitioning from one state to another depends only on the current state. It is a stochastic process that follows the Markov property, which states that the future behavior of the system depends only on its current state and is independent of its past history.
Here are some key points about Markov chains:
States: A Markov chain consists of a set of states, representing the possible conditions or configurations of the system. Each state is a distinct entity that the system can occupy at any given time.
Transitions: Transitions occur between states based on probabilities. At each step, the system moves from the current state to the next state according to transition probabilities. The transition probabilities capture the likelihood of moving from one state to another and are typically represented by a transition matrix or transition probability matrix.
Markov property: The Markov property implies that the probability of transitioning to the next state depends solely on the current state and is independent of the past states. In other words, the future behavior of the system is memoryless, and only the current state matters in determining the probabilities.
Transition probabilities: The transition probabilities are often represented by a square matrix, called the transition matrix. Each element of the matrix represents the probability of transitioning from one state to another. The sum of the probabilities in each row of the matrix is equal to 1.
Stationary distribution: A Markov chain may reach a state where the probabilities of transitioning between states no longer change. This state is called the stationary distribution or equilibrium distribution. In the stationary distribution, the probabilities of being in each state remain constant over time.
Applications: Markov chains find applications in various fields, including physics, biology, economics, computer science, and natural language processing. Some common applications include:
Weather prediction: Markov chains can be used to model and predict weather patterns based on the current weather conditions.
Stock market analysis: Markov chains can be applied to analyze and predict stock market movements based on historical data.
Natural language processing: Markov models can be used for tasks like text generation, speech recognition, and language modeling.
PageRank algorithm: Google’s PageRank algorithm, used for ranking web pages, is based on the idea of a Markov chain where web pages are represented as states, and transitions are based on hyperlinks.
Markov chains provide a probabilistic framework for modeling and analyzing systems that exhibit a memoryless property. They enable the study of the long-term behavior of a system and can be used to make predictions or simulations based on transition probabilities.
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