"A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event."
A stochastic model that describes a sequence of events where the probability of each event depends only on the state attained in the previous event.
Probability Theory: The foundation of Markov Chain analysis, probability theory helps us to understand the likelihood of certain events happening.
Stochastic Processes: Stochastic processes refer to a collection of random variables that evolve over time and are used to model complex systems. Markov chains fall under this category.
Random Walk: A random walk is a mathematical model that describes the motion of a random particle over time. Markov Chains can be represented as a random walk, where the future state of the system only depends on the current state.
Transition Probability: Transition probability is a mathematical function that describes the probability of moving from one state to another in a Markov Chain.
Stationary Distributions: A stationary distribution is an equilibrium state that a Markov Chain tends to converge to over time. It represents a steady-state distribution where the probability of being in each state remains constant.
Markov Chain Monte Carlo: Markov Chain Monte Carlo is a computational method used to simulate probability distributions by generating many random samples. It works by using a Markov Chain to move between different states to create a representative sample.
Hidden Markov Models: Hidden Markov Models are a type of statistical model that uses a Markov Chain to model a system where we don't have complete information about the state of the system. They are commonly used in speech recognition, hand-writing recognition, and bioinformatics.
Chapman-Kolmogorov Equation: The Chapman-Kolmogorov equation is a famous result in probability theory that describes the probability of transitioning from one state to another via an intermediate state in a Markov Chain.
Matrix Representation: Markov Chains can be represented as matrices, where each entry represents the transition probability from one state to another.
Ergodicity: Ergodicity refers to the property of Markov Chains that allows us to analyze the system's long-term behavior without having to simulate it for an infinitely long time. It guarantees that under certain conditions, the Markov Chain's behavior is independent of the starting state.
Discrete-Time Markov Chain: A Markov chain where the state transition probabilities are constant over time.
Continuous-Time Markov Chain: A Markov chain that is continuous in time, which means the probability of moving from one state to another is dependent upon the length of time.
Homogeneous Markov Chain: A Markov chain where the transition probabilities do not change over time.
Non-Homogeneous Markov Chain: A Markov chain where the transition probabilities change over time.
Irreducible Markov Chain: A Markov chain in which it is possible to transition from any state to any other state.
Reducible Markov Chain: A Markov chain in which it is not possible to transition from one or more states to one or more other states.
Ergodic Markov Chain: A Markov chain in which all the states are recurrent and positive recurrent.
Non-Ergodic Markov Chain: A Markov chain in which one or more states are transient.
Reversible Markov Chain: A Markov chain that satisfies the detailed balance condition, which means that the probability of going from state i to state j is equal to the probability of going from state j to state i.
Non-Reversible Markov Chain: A Markov chain that does not satisfy the detailed balance condition.
"What happens next depends only on the state of affairs now."
"A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC)."
"A continuous-time process is called a continuous-time Markov chain (CTMC)."
"It is named after the Russian mathematician Andrey Markov."
"Studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates, and animal population dynamics."
"Markov chains have many applications as statistical models of real-world processes."
"For simulating sampling from complex probability distributions."
"Bayesian statistics, thermodynamics, statistical mechanics, physics, chemistry, economics, finance, signal processing, information theory, and speech processing."
"The adjectives Markovian and Markov are used to describe something that is related to a Markov process."
"Studying cruise control systems in motor vehicles..."
"Animal population dynamics."
"Markov chain Monte Carlo (MCMC) methods."
"Bayesian statistics."
"Thermodynamics, statistical mechanics, physics, chemistry..."
"Economics, finance..."
"Signal processing, information theory..."
"Speech processing."
"The probability of each event depends only on the state attained in the previous event."
"Queues or lines of customers arriving at an airport."