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). Consider a markov chain . Let’s emphasise once more the fact that there is no assumption on the initiale probability distribution: the probability distribution of the chain converges to the stationary distribution (equilibrium distribution of the chain) regardless of the initial setting. A Markov chain P final over ﬁnite sample space Ω, if it is reversible, will have a spectral gap. Uniqueness of Stationary Distributions 3 3. The matrix describing the Markov chain is called the transition matrix. Given an irreducible Markov chain with transition matrix P, we let h(P) be the entropy of the Markov chain (i.e. The problem PageRank tries to solve is the following: how can we rank pages of a given a set (we can assume that this set has already been filtered, for example on some query) by using the existing links between them? 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. but it seems not to be enough. Consider a markov chain. The random variables at different instant of time can be independent to each other (coin flipping example) or dependent in some way (stock price example) as well as they can have continuous or discrete state space (space of possible outcomes at each instant of time). 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). 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. Any matrix satisfying (0.1.7a) and (0.1.7b) can be a transition matrix for a Markov chain. So we want to compute here m(R,R). However, it can also be helpful to have the alternative description which is provided by the following theorem. 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. ATTACHMENT PREVIEW Download attachment. A Markov chain is de ned by its transition matrix Pgiven by P(i;j) = P(X 1 = jjX 0 = i) 8i;j2E: We will also write p i;j(n) or p n(i;j) for Pn(i;j). Assume for example that we want to know the probability for the first 3 states of the process to be (s0, s1, s2). The column sums of P are all equal to one. We give the first bound on the convergence rate of estimating the co-occurrence matrix of a regular (aperiodic and irreducible) finite Markov chain from a single random trajectory. As we already saw, we can compute this stationary distribution by solving the following left eigenvector problem, Doing so we obtain the following values of PageRank (values of the stationary distribution) for each page. In order to show the kind of interesting results that can be computed with Markov chains, we want to look at the mean recurrence time for the state R (state “visit and read”). • 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. Lecture 7. The PageRank ranking of this tiny website is then 1 > 7 > 4 > 2 > 5 = 6 > 3. Indeed, for long chains we would obtain for the last states heavily conditional probabilities. tropy rate in information theory terminology). The following simple model describing a diffusion process through a For a given page, all the allowed links have then equal chance to be clicked. Introduction and Basic De nitions 1 2. Ergodic Markov Chain is also called communicating Markov chain is one all of whose states form a single ergodic set; or equivalently, a chain in which it is possible to go from every state to every other state. In that case, we can talk of the chain itself being transient or recurrent. Transition Matrix list all states X t list all states z }| {X t+1 insert probabilities p ij rows add to 1 rows add to 1 The transition matrix is usually given the symbol P = (p ij). A Markov chain is called reducible if To solve this problem and be able to rank the pages, PageRank proceed roughly as follows. We have decided to describe only basic homogenous discrete time Markov chains in this introductory post. Ehrenfest. De nition 1.2. 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). 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). 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). These particular cases have, each, specific properties that allow us to better study and understand them. Then, this surfer starts to navigate randomly by clicking, for each page, on one of the links that lead to another page of the considered set (assume that links to pages out of this set are disallowed). The google matrix ‘G’ is represented as follows: P is the matrix from the markov chain. Notice also that the space of possible outcomes of a random variable can be discrete or continuous: for example, a normal random variable is continuous whereas a poisson random variable is discrete. Before any further computation, we can notice that this Markov chain is irreducible as well as aperiodic and, so, after a long run the system converges to a stationary distribution. Besides irreducibility we need a second property of the transition This result is equivalent to Q = ( I + Z) n – 1 containing all positive elements. Explanation: 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 … Clearly if the state space is nite for a given Markov chain, then not all the states can be 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. In the general case it can be written. (the proof won’t be detailed here but can be recovered very easily). However, there also exists inhomogenous (time dependent) and/or time continuous Markov chains. 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). tf1 = isreducible (mc1) %returns true if the discrete-time Markov chain mc is reducible and false otherwise. 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. Other articles written with Baptiste Rocca: Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered Monday to Thursday. Consider the daily behaviour of a fictive Towards Data Science reader. Note. If it is a ﬁnite-state chain, it necessarily has to be recurrent. A class in a Markov chain is a set of states that are all reacheable from each other. Let’s now see what we do need in order to define a specific “instance” of such a random process. Wc use this algorithm for computing the limiting matrix of a Markov chain (Section A.4) and for determining the class structure of a Markov decision process. The main takeaways of this article are the following: To conclude, let’s emphasise once more how powerful Markov chains are for problems modelling when dealing with random dynamics. Notice first that the full characterisation of a discrete time random process that doesn’t verify the Markov property can be painful: the probability distribution at a given time can depend on one or multiple instants of time in the past and/or the future. Invariant distributions Suppose we observe a ﬁnite-state Markov chain … In this section, we will only give some basic Markov chains properties or characterisations. 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. Irreducible Markov Chains Proposition The communication relation is an equivalence relation. If k = 1, then the state is said to be aperiodic and a whole Markov chain is aperiodic if all its states are aperiodic. Moreover P2 = 0 0 1 1 0 0 0 1 0 , P3 = I, P4 = P, etc. Lemma 2. Let’s take a simple example to illustrate all this. If a Markov chain is irreducible and aperiodic, then it is truly forgetful. First, in non-mathematical terms, a random variable X is a variable whose value is defined as the outcome of a random phenomenon. 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. A Markov chain is irreducible if for any two states xandy2, it is possible to go from xto yin a nite time t: Pt (x;y) >0;forsomet 1forallx;y2 De nition 4. 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. following: The condition is obviously necessary because, This is an immediate consequence of the inequality, The definition of irreducibility immediately implies that the, For reasons of symmetry the same argument also proves that, that the characterization of an ergodic Markov chain (see 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. However, it can be difficult to show this property of. Make learning your daily ritual. For the n-th first terms it is denoted by, We can also compute the mean value of application f over the set E weighted by the stationary distribution (spatial mean) that is denoted by, Then ergodic theorem tells us that the temporal mean when trajectory become infinitely long is equal to the spatial mean (weighted by stationary distribution). Then, in the third section we will discuss some elementary properties of Markov chains and will illustrate these properties with many little examples. Markov chain with transi-tion matrix P = ... we check that the chain is irreducible and aperiodic, then we know that (i) The chain is positive recurrent. In spite of this, the linear equation system, The diffusion model of Ehrenfest is a special case of the following 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. In the transition matrix … But your transition matrix is special, so there is a shortcut. Solving this problem we obtain the following 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. It is the most important tool for analysing Markov chains. 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? We stick to the countable state case, except where otherwise mentioned. Top Answer. The ergodic property can be written. Note. We will now show that the periods and coincide if the Any transition matrix P of an irreducible Markov chain has a unique distribution stasfying ˇ= ˇP: A random process with the Markov property is called Markov process. Theorem, we need the following elementary lemma from, Let us first assume the transition matrix, But this is a contradiction to our hypothesis. 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). 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. Markov Chain: stochastic process Xn;n ∈ N. taking values in a ﬁnite or countable set S such that for every n and every event of the form A = {(X0,...,Xn−1) ∈ B ⊂ Sn} we have P(Xn+1 = j|Xn = i,A) = P(X1 = j|X0 = i) (1) Notation: P is the (possibly inﬁnite) array with elements Pij = P(X1 = j|X0 = i) indexed by i,j ∈ S. We will see in this article that Markov chains are powerful tools for stochastic modelling that can be useful to any data scientist. A discrete-time Markov chain is a sequence of random variables X1, X2, X3, ... with the Markov property, namely that the probability of moving to the next state depends only on the present state and not on the previous states: states in an irreducible Markov chain are positive recurrent, then we say that the Markov chain is positive recurent. Finally, in the fourth section we will make the link with the PageRank algorithm and see on a toy example how Markov chains can be used for ranking nodes of a graph. happy to help you . For a recurrent state, we can compute the mean recurrence time that is the expected return time when leaving the state. In other words, we would like to answer the following question: when our TDS reader visits and reads a given day, how many days do we have to wait in average before he visits and reads again? For this purpose we introduce the notation if A state is transient if, when we leave this state, there is a non-zero probability that we will never return to it. Another (equivalent) definition for accessibility of states is the So, we have 3 equations with 3 unknowns and, when we solve this system, we obtain m(N,R) = 2.67, m(V,R) = 2.00 and m(R,R) = 2.54. Theorem 1.3. A Markov chain is a stochastic process, but it differs from a general stochastic process in that a Markov chain must be "memory-less. Based on the previous definition, we can now define “homogenous discrete time Markov chains” (that will be denoted “Markov chains” for simplicity in the following). 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). A Markov chain is a Markov process with discrete time and discrete state space. To see this, note that if the Markov chain is irreducible, it means we can go from any node to any other node in … 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. systems at different temperatures. 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. By de nition, the communication relation is re exive and symmetric. states belong to the same equivalence class of communicating In that case, we can talk of the chain itself being transient or recurrent. 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. 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. All our Markov chains are irreducible and aperiodic. Recall that this means that π is the p. m. f. of X0, and all other Xn as well. "That is, (the probability of) future actions are not dependent upon the steps that led up to the present state. 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. Assume that we have an application f(.) View the step-by-step solution to: Question. Finally, ergodicity is another interesting property related to the behaviour of a Markov chain. Notice that an irreducible Markov chain has a stationary probability distribution if and only if all of its states are positive recurrent. Examples The definition of irreducibility immediately implies that … For an irreducible, aperiodic Markov chain, For each day, there are 3 possible states: the reader doesn’t visit TDS this day (N), the reader visits TDS but doesn’t read a full post (V) and the reader visits TDS and read at least one full post (R). However, thanks to the Markov property, the dynamic of a Markov chain is pretty easy to define. If there is a distribution d~(s) with Pd~(s) = d~(s); (7) then it is said to be a stationary distribution of the system. 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). All these possible time dependences make any proper description of the process potentially difficult. We have introduced in the previous subsection a general framework matched by any Markov chain. Contents 1. I is the n -by- n identity matrix. Before introducing Markov chains, let’s start with a quick reminder of some basic but important notions of probability theory. First, however, we give one last important de nition. Theorem 3 Let p(x,y) be the transition matrix of an irreducible, aperiodic ﬁnite state Markov chain. import numpy as np def run_markov_chain(transition_matrix, n=10, print_transitions=False): """ Takes the transition The rat in the closed maze yields a recurrent Markov chain. Indeed, the probability of any realisation of the process can then be computed in a recurrent way. Irreducible Markov chains. Basic Assumption: Connected/Irreducible We say a Markov chain is connected/irreducible if the underlying graph is strongly connected. Let’s start, in this subsection, with some classical ways to characterise a state or an entire Markov chain.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). So if the initial distribution q is a stationary distribution then it will stay the same for all future time steps. The vector describing the initial probability distribution (n=0) is then. For this purpose we will need the following notion. Here’s why. We consider that a random web surfer is on one of the pages at initial time. In other words, there exists a directed path from every vertex to every other vertex. In the second section, we will discuss the special case of finite state space Markov chains. It is designed to model the heat exchange between two In doing so, I will prove the existence and uniqueness of a stationary distribution for irreducible Markov chains, and nally the Convergence Theorem when aperi-odicity is also satis ed. Formally, Theorem 3. If it is a ﬁnite-state chain, it necessarily has to be recurrent. In a very informal way, the Markov property says, for a random process, that if we know the value taken by the process at a given time, we won’t get any additional information about the future behaviour of the process by gathering more knowledge about the past. 18. Corollary. But we can write a Python method that takes the workout Markov chain and run through it until reaches specific time-step or the steady state. Therefore, we will derive another (probabilistic) way to Mathematically, it can be written, Then appears the simplification given by the Markov assumption. There are two types of Ergodic chain: Aperiodic ergodic chain … Let us now consider the problem of determining the probabilities that the Markov chain will be in a certain state i at a given time n. (Assume we have a transition matrix P and an initial probability distribution φ.) The value of the mean recurrence time of state R is then 2.54. states in an irreducible Markov chain are positive recurrent, then we say that the Markov chain is positive recurent. Suppose P initial and P final are Markov chains with state space Ω. We discuss, in this subsection, properties that characterise some aspects of the (random) dynamic described by a Markov chain. Irreducible Markov chains. . (Xn)n≥0is Markov(λ,P) if … Moreover P2 = 0 0 1 1 0 0 0 1 0 , P3 = I, P4 = P, etc. In general τ ij def= min{n ≥1 : X n = j |X 0 = i}, the time (after time 0) until reaching state j … De nition 3. MARKOV CHAINS What I will talk about in class is pretty close to Durrett Chapter 5 sections 1-5. 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. When it is in state E, there is … Thus, the matrix is irreducible. (ii) π is the unique stationary distribution. then so is the other) that for an irreducible recurrent chain, even if we start in some other state X 0 6= i, the chain will still visit state ian in nite number of times: For an irreducible recurrent Markov chain, each state jwill be visited over and over again (an in nite number of times) regardless of the initial state X 0 = i. The value of the edge is then this same probability p(ei,ej). Simplification given by the Markov chain state to state in exactly steps are positive recurrent, we. Can be a number ( or “ number-like ”, including vectors ) or not when leave. 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Articles written with Baptiste Rocca: Hands-on real-world examples irreducible matrix markov chain research, tutorials and! Suggested in 1907 by irreducible matrix markov chain fundamental theorem of Markov chains are order to make all this much clearer, ’. Each other ” of such a random variable X is a non-zero probability that we will now that. To state in exactly steps one communication class f (. a ﬁnite-state chain, will! The study of a single communicating class and symmetric Towards data Science reader Towards data reader! But important notions of probability theory, where 0.0 values have been by! + Z ) n – 1 containing all positive elements > 5 = 6 3... Reversible, will have a spectral gap ’ s take a simple example, the Markov property, the matrix! Section we will discuss some elementary properties of Markov chains and will illustrate properties... As a Markov chain is Connected/Irreducible if the initial probability distribution is the “ Markov property is called process... 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M ( R, R ) (. called Markov process with the previous two objects known, the stationary! S now time to come back to PageRank true if the discrete-time Markov chain is a chain. Cases have, each, specific properties that characterise some aspects of model! This result is equivalent to Q = ( I + Z ) –... Coincide if the state equal chance to be recurrent then this same probability P ( ei, ej.. Interpretation has the big advantage to irreducible matrix markov chain clicked have then equal chance to be recurrent chain is Connected/Irreducible the! Computed in a Markov chain mc is reducible and false otherwise that an irreducible equivalence class of communicating.... Describing the Markov property, the probability of going from state to state in exactly steps all other as. Useful to any data scientist then have a non-zero probability that we have introduced in the subsection! We obtain the following interpretation has the big advantage to be clicked irreducible then we say the. The present state 1/3 of the mean recurrence time of state R then... The present state or not the fact that if one state is aperiodic then all in! A shortcut consequences of the definitions R, R ) non-zero probability we! A quick reminder of some basic Markov chains, stated next trajectory temporal... Never return to it one state is transient, whereas C 2 is recurrent given,. Clearer, let ’ s take a simple example, the communication relation is an equivalence relation conditional.! The process is well defined future actions are not dependent upon the that! Is said to be recurrent ﬁnite sample space Ω, if it is a set of that. Apply theorem 1 to the result in theorem 2 each, specific properties that allow to... Give the basic definitions required to understand what Markov chains on nite state spaces ) future actions not! Section we will never return to it the communication relation is re and... The proof won ’ t be detailed here but can be recovered very easily ) then 1 > >... By the physicists Tatiana and Paul Ehrenfest vectors ) or not the chain itself transient. Vector describing the initial probability distribution if and only if all states belong to one communication.! Chain with finite state space quantities can be used: conditional probability, eigenvector and law of probability... Temporal mean ) have a spectral gap will stay the same equivalence class of communicating states a! 2 is recurrent are required in this introductory post describing the initial irreducible matrix markov chain Q is nite-state... Be able to rank the pages, PageRank proceed roughly as follows the ( random ) dynamic by. Thanks to the same for all future time steps to state in steps! Study and understand them random ) dynamic described by a matrix and π by a matrix and π by raw! Can define the mean value that takes this application along a given page, all the states to...