Due to their good properties, they are used in various fields such as queueing theory (optimising the performance of telecommunications networks, where messages must often compete for limited resources and are queued when all ressources are already allocated), statistics (the well known “Markov Chain Monte Carlo” random variables generation technique is based on Markov chains), biology (modelling of biological populations evolution), computer science (hidden Markov models are important tools in information theory and speech recognition) and others. Once more, it expresses the fact that a stationary probability distribution doesn’t evolve through the time (as we saw that right multiplying a probability distribution by p allows to compute the probability distribution at the next time step). View the step-by-step solution to: Question. A Markov chain is a Markov process with discrete time and discrete state space. To conclude this example, let’s see what the stationary distribution of this Markov chain is. For this purpose we introduce the notation if Note. If it is a nite-state chain, it necessarily has to be recurrent. Theorem 3 Let p(x,y) be the transition matrix of an irreducible, aperiodic ﬁnite state Markov chain. Imagine also that the following probabilities have been observed: Then, we have the following transition matrix, Based on the previous subsection, we know how to compute, for this reader, the probability of each state for the second day (n=1), Finally, the probabilistic dynamic of this Markov chain can be graphically represented as follows. Finally, the Markov chain is said to be irreducible it it consists of a single communicating class. A little bit more than two decades later, Google has became a giant and, even if the algorithm has evolved a lot, the PageRank is still a “symbol” of the Google ranking algorithm (even if few people can really say the weight it still occupies in the algorithm). Although the chain does spend 1/3 of the time at each state, the transition In probability, a Markov chain is a sequence of random variables, known as a stochastic process, in which the value of the next variable depends only on the value of the current variable, and not any variables in the past. If the state space is ﬁnite and all states communicate (that is, the Markov chain is irreducible) then in the long run, regardless of the initial condition, the Markov chain must settle into a steady state. for all . We can regard (p(i,j)) as deﬁning a (maybe inﬁnite) matrix P. Then a basic fact is P(X n = j|X0 = i)=Pn(i,j) (12) where Pn denotes matrix multiplication. In that case, we can talk of the chain itself being transient or recurrent. Finally, the Markov chain is said to be irreducible it it consists of a single communicating class. More especially, we will answer basic questions such as: what are Markov chains, what good properties do they have and what can be done with them? Consider a markov chain . Lemma 2. So, a Markov chain is a discrete sequence of states, each drawn from a discrete state space (finite or not), and that follows the Markov property. We won’t discuss these variants of the model in the following. but it seems not to be enough. Basics of probability and linear algebra are required in this post. Here we apply Theorem 1 to the result in Theorem 2. If we assume also that the defined chain is recurrent positive and aperiodic (some minor tricks are used to ensure we meet this setting), then after a long time the “current page” probability distribution converges to the stationary distribution. For example, flipping a coin every day defines a discrete time random process whereas the price of a stock market option varying continuously defines a continuous time random process. Formally, Theorem 3. Finally, ergodicity is another interesting property related to the behaviour of a Markov chain. In the first section we will give the basic definitions required to understand what Markov chains are. However, there also exists inhomogenous (time dependent) and/or time continuous Markov chains. • If there exists some n for which p ij (n) >0 for all i and j, then all states communicate and the Markov chain is irreducible. In the transition matrix … To better grasp that convergence property, let’s take a look at the following graphic that shows the evolution of probability distributions beginning at different starting point and the (quick) convergence to the stationary distribution. So, among the recurrent states, we can make a difference between positive recurrent state (finite expected return time) and null recurrent state (infinite expected return time). Besides irreducibility we need a second property of the transition If the chain is recurrent positive (so that there exists a stationary distribution) and aperiodic then, no matter what the initial probabilities are, the probability distribution of the chain converges when time steps goes to infinity: the chain is said to have a limiting distribution that is nothing else than the stationary distribution. These particular cases have, each, specific properties that allow us to better study and understand them. The google matrix ‘G’ is represented as follows: P is the matrix from the markov chain. Mathematically, we can denote a Markov chain by, where at each instant of time the process takes its values in a discrete set E such that, Then, the Markov property implies that we have. In this simple example, the chain is clearly irreducible, aperiodic and all the states are recurrent positive. If the state space is finite and the chain can be represented by a graph, then we can say that the graph of an irreducible Markov chain is strongly connected (graph theory). Let’s take a simple example to illustrate all this. Indeed, we only need to specify two things: an initial probability distribution (that is a probability distribution for the instant of time n=0) denoted, and a transition probability kernel (that gives the probabilities that a state, at time n+1, succeeds to another, at time n, for any pair of states) denoted. De nition 3. A simple example for a non-irreducible Markov chain, can be given by our well-known model for the, It is nevertheless possible that the linear equation system, Now we give some examples for non-aperiodic Markov chains, We merely assume that the random variables. Let’s try to get an intuition of how to compute this value. We have introduced in the previous subsection a general framework matched by any Markov chain. These two quantities can be expressed the same way. In particular, the following notions will be used: conditional probability, eigenvector and law of total probability. As we have seen that in the finite state space case we can picture a Markov chain as a graph, notice that we will use graphical representation to illustrate some of the properties bellow. For a recurrent state, we can compute the mean recurrence time that is the expected return time when leaving the state. 16 MARKOV CHAINS: REVERSIBILITY 182 16 Markov Chains: Reversibility Assume that you have an irreducible and positive recurrent chain, started at its unique invariant distribution π. h(P) = P i;j ˇ iP ijlogP ij where ˇ i is the (unique) invariant distribution of the Markov chain and where as usual … Apple’s New M1 Chip is a Machine Learning Beast, A Complete 52 Week Curriculum to Become a Data Scientist in 2021, How to Become Fluent in Multiple Programming Languages, 10 Must-Know Statistical Concepts for Data Scientists, How to create dashboard for free with Google Sheets and Chart.js, Pylance: The best Python extension for VS Code, when the reader doesn’t visit TDS a day, he has 25% chance of still not visiting the next day, 50% chance to only visit and 25% to visit and read, when the reader visits TDS without reading a day, he has 50% chance to visit again without reading the next day and 50% to visit and read, when the reader visits and read a day, he has 33% chance of not visiting the next day, random processes are collections of random variables, often indexed over time (indices often represent discrete or continuous time), for a random process, the Markov property says that, given the present, the probability of the future is independent of the past (this property is also called “memoryless property”), discrete time Markov chain are random processes with discrete time indices and that verify the Markov property, the Markov property of Markov chains makes the study of these processes much more tractable and allows to derive some interesting explicit results (mean recurrence time, stationary distribution…), one possible interpretation of the PageRank (not the only one) consists in imagining a web surfer that randomly navigates from page to page and in taking the induced stationary distribution over pages as a factor of ranking (roughly, the most visited pages in steady-state must be the one linked by other very visited pages and then must be the most relevant). A random process with the Markov property is called Markov process. Finding it difficult to learn programming? In that case, we can talk of the chain itself being transient or recurrent. The invariant probability π will be unique, since your chain is irreducible. MARKOV CHAINS What I will talk about in class is pretty close to Durrett Chapter 5 sections 1-5. Stated in slightly more mathematical terms, for any given time, the conditional distribution of future states of the process given present and past states depends only on the present state and not at all on the past states (memoryless property). If the state space is finite and the chain can be represented by a graph, then we can say that the graph of an irreducible Markov chain is strongly connected (graph theory). Invariant distributions Suppose we observe a ﬁnite-state Markov chain … A Markov chain is called irreducible if for all x;y2Ethere exists n 0 such that Pn(x;y) >0. For clarity the probabilities of each transition have not been displayed in the previous representation. states in an irreducible Markov chain are positive recurrent, then we say that the Markov chain is positive recurent. We have decided to describe only basic homogenous discrete time Markov chains in this introductory post. Assume that we have a tiny website with 7 pages labeled from 1 to 7 and with links between the pages as represented in the following graph. Basic Assumption: Connected/Irreducible We say a Markov chain is connected/irreducible if the underlying graph is strongly connected. Then for all states x,y, lim n→∞ pn(x,y) = π(y) (7.9) For any initial distribution πo, the distribution πn of Xn converges to the stationary distribution π. "That is, (the probability of) future actions are not dependent upon the steps that led up to the present state. Make learning your daily ritual. Notice also that the definition of the Markov property given above is extremely simplified: the true mathematical definition involves the notion of filtration that is far beyond the scope of this modest introduction. Stated in another way, no matter what the initial state of our TDS reader is, if we wait long enough and pick a day randomly then we have a probability π(N) that the reader doesn’t visit for this day, a probability π(V) that the reader visits but doesn’t read and a probability π(R) that the reader visits and reads. otherwise. Introduction and Basic De nitions 1 2. • If a Markov chain is not irreducible… If the state space is ﬁnite and all states communicate (that is, the Markov chain is irreducible) then in the long run, regardless of the initial condition, the Markov chain must settle into a steady state. Here’s why. Stated in another way, it says that, at the limit, the early behaviour of the trajectory becomes negligible and only the long run stationary behaviour really matter when computing the temporal mean. To determine the stationary distribution, we have to solve the following linear algebra equation, So, we have to find the left eigenvector of p associated to the eigenvalue 1. 15 MARKOV CHAINS: LIMITING PROBABILITIES 170 This is an irreducible chain, with invariant distribution π0 = π1 = π2 = 1 3 (as it is very easy to check). Another (equivalent) definition for accessibility of states is the Invariant distributions Suppose we observe a ﬁnite-state Markov chain … A state has period k if, when leaving it, any return to that state requires a multiple of k time steps (k is the greatest common divisor of all the possible return path length). However, the following interpretation has the big advantage to be very well understandable. Before going any further, let’s mention the fact that the interpretation that we are going to give for the PageRank is not the only one possible and that authors of the original paper had not necessarily in mind Markov chains when designing the method. Notice once again that this last formula expresses the fact that for a given historic (where I am now and where I was before), the probability distribution for the next state (where I go next) only depends on the current state and not on the past states. . Obviously, the huge possibilities offered by Markov chains in terms of modelling as well as in terms of computation go far behind what have been presented in this modest introduction and, so, we encourage the interested reader to read more about these tools that entirely have there place in the (data) scientist toolbox. Notice that even if the probability of return is equal to 1, it doesn’t mean that the expected return time is finite. (Xn)n≥0is Markov(λ,P) if … We have here a the setting of a Markov chain: pages are the different possible states, transition probabilities are defined by the links from page to page (weighted such that on each page all the linked pages have equal chances to be chosen) and the memoryless properties is clearly verified by the behaviour of the surfer. Assume that we have an application f(.) Co-occurrence statistics for sequential data are common and important data signals in machine learning, which provide rich correlation and clustering information about the underlying object space. Corollary. First, we say that a Markov chain is irreducible if it is possible to reach any state from any other state (not necessarily in a single time step). Other articles written with Baptiste Rocca: Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. A class in a Markov chain is a set of states that are all reacheable from each other. Irreducible Markov Chains Proposition The communication relation is an equivalence relation. Notice that an irreducible Markov chain has a stationary probability distribution if and only if all of its states are positive recurrent. But your transition matrix is special, so there is a shortcut. However, one should keep in mind that these properties are not necessarily limited to the finite state space case. In the second section, we will discuss the special case of finite state space Markov chains. The column sums of P are all equal to one. For example we can define a random variable as the outcome of rolling a dice (number) as well as the output of flipping a coin (not a number, unless you assign, for example, 0 to head and 1 to tail). Then came accros a part saying that the object should be defined first as a Markov chain. states belong to the same equivalence class of communicating However, in a Markov case we can simplify this expression using that, As they fully characterise the probabilistic dynamic of the process, many other more complex events can then be computed only based on both the initial probability distribution q0 and the transition probability kernel p. One last basic relation that deserves to be given is the expression of the probability distribution at time n+1 expressed relatively to the probability distribution at time n. We assume here that we have a finite number N of possible states in E: Then, the initial probability distribution can be described by a row vector q0 of size N and the transition probabilities can be described by a matrix p of size N by N such that, The advantage of such notation is that if we note denote the probability distribution at step n by a raw vector qn such that its components are given by, then the simple matrix relations thereafter hold.
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