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Markov Chain

Markov Chain is a probabilistic model for the analysis of random processes. It is named after Andrey Markov, a Russian mathematician who first introduced the concept in his work on probability theory in the early 20th century. A Markov Chain is a sequence of random variables that have a property known as the Markov property. This property states that the probability distribution of a future state of the system depends only on the current state and not on any of the past states.

Definition

Formally, a Markov Chain is a sequence of random variables X0,X1,X2,...{X_0, X_1, X_2, ...} that take values from a finite or countably infinite set called the state space SS. The Markov property can be expressed as follows:

P(Xn+1=j∣X0=i0,X1=i1,…,Xn=in)=P(Xn+1=j∣Xn=in)P(X_{n+1}=j | X_0=i_0,X_1=i_1, \dots, X_n=i_n) = P(X_{n+1}=j | X_n=i_n)

for any n≥0,i0,i1,…,in,j∈Sn \geq 0, i_0, i_1, \dots, i_n, j \in S. In other words, the probability of moving to the next state depends only on the current state and not on any of the previous states.

Transition Matrix

The behavior of a Markov Chain can be described by a matrix called the transition matrix. The transition matrix PP of a Markov Chain is an n×nn \times n matrix, where nn is the number of states in the state space. The entry PijP_{ij} represents the probability of transitioning from state ii to state jj. For any i∈Si \in S, the sum of the probabilities of transitioning from state ii to all other states is 1:

∑j∈SPij=1\sum_{j \in S} P_{ij} = 1

Stationary Distribution

A stationary distribution for a Markov Chain is a probability distribution that remains unchanged over time. In other words, if the system starts with a distribution that is equal to the stationary distribution, then the distribution will remain unchanged as the system evolves over time. Mathematically, a stationary distribution π\pi satisfies the following equation:

πj=∑i∈SπiPij\pi_j = \sum_{i \in S} \pi_i P_{ij}

for all j∈Sj \in S. This equation is known as the balance equation or the detailed balance equation.

Applications

Markov Chains have many applications in various fields such as physics, chemistry, economics, and computer science. In physics, Markov Chains are used to model the behavior of particles in a system. In chemistry, Markov Chains are used to model the behavior of molecules in a solution. In economics, Markov Chains are used to model the behavior of financial markets. In computer science, Markov Chains are used in various applications such as natural language processing, speech recognition, and image processing.

Conclusion

Markov Chains are a powerful tool for modeling complex systems that exhibit random behavior. They have numerous applications in various fields and are widely used in research and industry. Understanding the properties of Markov Chains and their applications is essential for anyone interested in the field of probability theory and its applications.

マルコフ連鎖[JA]