Lecture 7: Markov Chains and Random Walks
نویسندگان
چکیده
A transition probability Pij corresponds to the probability that the state at time step t+1 will be j, given that the state at time t is i. Therefore, each row in the matrix M is a distribution and ∀i, j ∈ SPij ≥ 0 and ∑ j Pij = 1. Let the initial distribution be given by the row vector x ∈ R, xi ≥ 0 and ∑ i xi = 1. After one step, the new distribution is xM. It is easy to see that xM is again a distribution. Sometimes it is useful to think of x as describing a certain amount fluid sitting at each node, such that the sum of the amounts is 1. After one step, the fluid sitting at node i distributes to its neighbors, such that Pij fraction goes to j. We stress that the evolution of a Markov chain is memoryless: the transition probability Pij depends only on the state i and not on the time t or the sequence of transititions taken before this time. Suppose we take two steps in this Markov chain. The memoryless property implies that the probability of going from i to j is ∑ k PikPkj , which is just the (i, j)th entry of the matrix M2. In general taking t steps in the Markov chain corresponds to the matrix M t.
منابع مشابه
Random Walks on Finite Quantum Groups
1 Markov chains and random walks in classical probability . . 3 2 Quantum Markov chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 Random walks on comodule algebras . . . . . . . . . . . . . . . . . . . . . . 7 4 Random walks on finite quantum groups . . . . . . . . . . . . . . . . . . 11 5 Spatial Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . ....
متن کاملRandom walks and Markov chains
This lecture discusses Markov chains, which capture and formalize the idea of a memoryless random walk on a finite number of states, and which have wide applicability as a statistical model of many phenomena. Markov chains are postulated to have a set of possible states, and to transition randomly from one state to a next state, where the probability of transitioning to a particular next state ...
متن کاملRandom Walks on Infinite Graphs and Groups — a Survey on Selected Topics
Contents 1. Introduction 2 2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4 C. Random walks on groups 5 D. Group-invariant random walks on graphs 6 E. Harmonic and superharmonic functions 6 3. Spectral radius, amenability and law of large numbers 6 A. Spectral radius, isoperimetric inequalities and growth 6 B. Law of large numbers 9 ...
متن کاملRandom Walks and Brownian Motion Instructor :
In this lecture we compute asumptotics estimates for the Green's function and apply it to the exiting annuli problem. Also we define Capacity, Polar set Prove the B-P-P theorem of Martin's Capacity for Markov chains[3] and use apply it on the intersection of RW problem.
متن کاملLecture 14 : Random Walks
Markov Chains have many applications in different areas of science including computer science, mathematics, finance, economics, etc. Let us describe an application in speech recognition. One of the important tasks in natural language processing is to predict the next word of a sentence given the past words. One way to model this problem is by a Markov chain. Say we have a chain where each state...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2005