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. Informally, this may be thought of as, "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.
Markov chains have many applications as statistical models of real-world processes, such as studying cruise control systems in motor vehicles, queues or lines of customers arriving at an airport, currency exchange rates and animal population dynamics.
Markov processes are the basis for general stochastic simulation methods known as Markov chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found application in 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.
A Markov process is a stochastic process that satisfies the Markov property (sometimes characterized as "memorylessness"). In simpler terms, it is a process for which predictions can be made regarding future outcomes based solely on its present state and—most importantly—such predictions are just as good as the ones that could be made knowing the process's full history. In other words, conditional on the present state of the system, its future and past states are independent.
A Markov chain is a type of Markov process that has either a discrete state space or a discrete index set (often representing time), but the precise definition of a Markov chain varies.
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Internet analytics is the collection, modeling, and analysis of user data in large-scale online services, such as social networking, e-commerce, search, and advertisement. This class explores a number
The goal of this class is to acquire mathematical tools and engineering insight about networks whose structure is random, as well as learning and control techniques applicable to such network data.
Information is processed in physical devices. In the quantum regime the concept of classical bit is replaced by the quantum bit. We introduce quantum principles, and then quantum communications, key d
Andrey Andreyevich Markov (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as the Markov chain. He was also a strong, close to master-level chess player. Markov and his younger brother Vladimir Andreevich Markov (1871–1897) proved the Markov brothers' inequality. His son, another Andrey Andreyevich Markov (1903–1979), was also a notable mathematician, making contributions to constructive mathematics and recursive function theory.
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. Informally, this may be thought of as, "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).
In probability theory and related fields, a stochastic (stəˈkæstɪk) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.
This thesis is devoted to the construction, analysis, and implementation of two types of hierarchical Markov Chain Monte Carlo (MCMC) methods for the solution of large-scale Bayesian Inverse Problems
We study in this thesis the asymptotic behavior of optimal paths on a random graph model, the configuration model, for which we assign continuous random positive weights on its edges.
We start by desc
In this work, we present, analyze, and implement a class of Multi-Level Markov chain Monte Carlo (ML-MCMC) algorithms based on independent Metropolis-Hastings proposals for Bayesian inverse problems.