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Lecture

Markov Chains: Reversibility & Convergence

In course

MATH-414: Stochastic simulation

The student who follows this course will get acquainted with computational tools used to analyze systems with uncertainty arising in engineering, physics, chemistry, and economics. Focus will be on sampling methods as Monte Carlo, quasi Monte Carlo, Mark ...

Description

This lecture covers the concepts of Markov chains, focusing on reversibility and convergence. Topics include discrete state Markov chains, Metropolis-Hastings algorithm, and the general state space of Markov chains. The instructor explains the properties of reversible Markov chains and discusses the exponential convergence of Markov chains. Additionally, the lecture delves into ergodicity, geometric ergodicity, and the convergence of Markov chains. The presentation concludes with detailed explanations of the convergence properties of Markov chains and their applications.

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Instructor

Fabio Nobile

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Related concepts (32)

Uniform convergence

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set as the function domain if, given any arbitrarily small positive number , a number can be found such that each of the functions differs from by no more than at every point in .

Ergodicity

In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process.

Ergodic theory

Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc.

Web application

A web application (or web app) is application software that is accessed using a web browser. Web applications are delivered on the World Wide Web to users with an active network connection. In earlier computing models like client-server, the processing load for the application was shared between code on the server and code installed on each client locally. In other words, an application had its own pre-compiled client program which served as its user interface and had to be separately installed on each user's personal computer.

Ergodic hypothesis

In physics and thermodynamics, the ergodic hypothesis says that, over long periods of time, the time spent by a system in some region of the phase space of microstates with the same energy is proportional to the volume of this region, i.e., that all accessible microstates are equiprobable over a long period of time. Liouville's theorem states that, for a Hamiltonian system, the local density of microstates following a particle path through phase space is constant as viewed by an observer moving with the ensemble (i.

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