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. Thus, the matrix is irreducible. We discuss, in this subsection, properties that characterise some aspects of the (random) dynamic described by a Markov chain. The hypothesis behind PageRank is that the most probable pages in the stationary distribution must also be the most important (we visit these pages often because they receive links from pages that are also visited a lot in the process). If the state space is finite, p can be represented by a matrix and π by a raw vector and we then have. states. All these possible time dependences make any proper description of the process potentially difficult. 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. Indeed, the probability of any realisation of the process can then be computed in a recurrent way. Such a transition matrix is called doubly stochastic and its unique invariant probability measure is uniform, i.e., π = … The rat in the closed maze yields a recurrent Markov chain. If all states in an irreducible Markov chain are null recurrent, then we say that the Markov chain is null recurent. Conversely, the irreducibility and aperiodicity of quasi-positive Basic Assumption: Connected/Irreducible We say a Markov chain is connected/irreducible if the underlying graph is strongly connected. Then, in the third section we will discuss some elementary properties of Markov chains and will illustrate these properties with many little examples. Introduction and Basic De nitions 1 2. Top Answer. (iii) π is the limiting distribution. import numpy as np def run_markov_chain(transition_matrix, n=10, print_transitions=False): """ Takes the transition In general τ ij def= min{n ≥1 : X n = j |X 0 = i}, the time (after time 0) until reaching state j … Notice that even if the probability of return is equal to 1, it doesn’t mean that the expected return time is finite. There are two types of Ergodic chain: Aperiodic ergodic chain … We will now show that the periods and coincide if the 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). Uniqueness of Stationary Distributions 3 3. In the first section we will give the basic definitions required to understand what Markov chains are. 2. Theorem 3 Let p(x,y) be the transition matrix of an irreducible, aperiodic ﬁnite state Markov chain. Notice that an irreducible Markov chain has a stationary probability distribution if and only if all of its states are positive recurrent. The value of the edge is then this same probability p(ei,ej). to characterize the ergodicity of a Markov chain in a simple way. It is designed to model the heat exchange between two Indeed, for long chains we would obtain for the last states heavily conditional probabilities. but it seems not to be enough. We will see in this article that Markov chains are powerful tools for stochastic modelling that can be useful to any data scientist. 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). Before introducing Markov chains, let’s start with a quick reminder of some basic but important notions of probability theory. In that case, we can talk of the chain itself being transient or recurrent. 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”). transition matrices are immediate consequences of the definitions. Consider a markov chain . 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). Markov Chains - 10 Irreducibility • A Markov chain is irreducible if all states belong to one class (all states communicate with each other). We have decided to describe only basic homogenous discrete time Markov chains in this introductory post. 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. For an irreducible, aperiodic Markov chain, The google matrix ‘G’ is represented as follows: P is the matrix from the markov chain. IMG-20201217-WA0060.jpg. Given an irreducible Markov chain with transition matrix P, we let h(P) be the entropy of the Markov chain (i.e. 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). 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. 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. 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. In other words, there exists a directed path from every vertex to every other vertex. 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. Easy Handling Discrete Time Markov Chains: rctmc: rctmc: names,markovchain-method: Returns the states for a Markov chain object: rmarkovchain: Function to generate a sequence of states from homogeneous or non-homogeneous Markov chains. 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 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. 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). A probability distribution π over the state space E is said to be a stationary distribution if it verifies, By definition, a stationary probability distribution is then such that it doesn’t evolve through the time. A probability distribution ˇis stationary for a Markov chain with transition matrix P if ˇP= ˇ. Irreducible Markov chains. Explanation: In this simple example, the chain is clearly irreducible, aperiodic and all the states are recurrent positive. For an irreducible Markov chain, we can also mention the fact that if one state is aperiodic then all states are aperiodic. Corollary. These particular cases have, each, specific properties that allow us to better study and understand them. Thanks a lot! 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. However, as the “navigation” is supposed to be purely random (we also talk about “random walk”), the values can be easily recovered using the simple following rule: for a node with K outlinks (a page with K links to other pages), the probability of each outlink is equal to 1/K. In order to make all this much clearer, let’s consider a toy example. If k = 1, then the state is said to be aperiodic and a whole Markov chain is aperiodic if all its states are aperiodic. 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? Finally, the Markov chain is said to be irreducible it it consists of a single communicating class. Here’s why. In the transition matrix … It is the most important tool for analysing Markov chains. A Markov chain is called irreducible if for all x;y2Ethere exists n 0 such that Pn(x;y) >0. We can define the mean value that takes this application along a given trajectory (temporal mean). The invariant probability π will be unique, since your chain is irreducible. 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? If all states in an irreducible Markov chain are null recurrent, then we say that the Markov chain is null recurent. If the Markov chain is irreducible and aperiodic, then the Markov chain is primitive (such that ). More generally, suppose that \( \bs{X} \) is a Markov chain with state space \( S \) and transition probability matrix \( P \). Here we apply Theorem 1 to the result in Theorem 2. Examples The definition of irreducibility immediately implies that … 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. The PageRank ranking of this tiny website is then 1 > 7 > 4 > 2 > 5 = 6 > 3. As the chain is irreducible and aperiodic, it means that, in the long run, the probability distribution will converge to the stationary distribution (for any initialisation). 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. Conversely, a state is recurrent if we know that we will return to that state, in the future, with probability 1 after leaving it (if it is not transient). For clarity the probabilities of each transition have not been displayed in the previous representation. Checking conditions (i) and (ii) is usually the most helpful way to determine whether or not a given random process (Xn)n≥0is a Markov chain. View the step-by-step solution to: Question. In that case, we can talk of the chain itself being transient or recurrent. 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. The random dynamic of a finite state space Markov chain can easily be represented as a valuated oriented graph such that each node in the graph is a state and, for all pairs of states (ei, ej), there exists an edge going from ei to ej if p(ei,ej)>0. closed irreducible classes and transient states of a finite Markov chain. Although the chain does spend 1/3 of the time at each state, the transition Invariant distributions Suppose we observe a ﬁnite-state Markov chain … To solve this problem and be able to rank the pages, PageRank proceed roughly as follows. MARKOV CHAINS What I will talk about in class is pretty close to Durrett Chapter 5 sections 1-5. 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: Therefore, we will derive another (probabilistic) way to otherwise. The rat in the open maze yields a Markov chain that is not irreducible; there are two communication classes C 1 = f1;2;3;4g;C 2 = f0g. We can then define a random process (also called stochastic process) as a collection of random variables indexed by a set T that often represent different instants of time (we will assume that in the following). For a recurrent state, we can compute the mean recurrence time that is the expected return time when leaving the state. Assume for example that we want to know the probability for the first 3 states of the process to be (s0, s1, s2). With the previous two objects known, the full (probabilistic) dynamic of the process is well defined. 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 Another interesting property related to stationary probability distribution is the following. 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). In 1998, Lawrence Page, Sergey Brin, Rajeev Motwani and Terry Winograd published “The PageRank Citation Ranking: Bringing Order to the Web”, an article in which they introduced the now famous PageRank algorithm at the origin of Google. 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. 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. 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. Finally, the Markov chain is said to be irreducible it it consists of a single communicating class. For a given page, all the allowed links have then equal chance to be clicked. These two quantities can be expressed the same way. 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. Invariant distributions Suppose we observe a ﬁnite-state Markov chain … (ii) π is the unique stationary distribution. Contents 1. Moreover P2 = 0 0 1 1 0 0 0 1 0 , P3 = I, P4 = P, etc. However, thanks to the Markov property, the dynamic of a Markov chain is pretty easy to define. states in an irreducible Markov chain are positive recurrent, then we say that the Markov chain is positive recurent. 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. Moreover P2 = 0 0 1 1 0 0 0 1 0 , P3 = I, P4 = P, etc. However, there also exists inhomogenous (time dependent) and/or time continuous Markov chains. An irreducible 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. 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. . happy to help you . 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. Although the chain does spend 1/3 of the time at each state, 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. First, however, we give one last important de nition. So, we have the following state space, Assume that at the first day this reader has 50% chance to only visit TDS and 50% chance to visit TDS and read at least one article. In the general case it can be written. From a theoretical point of view, it is interesting to notice that one common interpretation of the PageRank algorithm relies on the simple but fundamental mathematical notion of Markov chains. 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). For this purpose we introduce the notation if De nition 1.2. 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). 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. The column sums of P are all equal to one. 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. tropy rate in information theory terminology). Irreducible Markov Chains Proposition The communication relation is an equivalence relation. Consider a markov chain. The matrix ) is called the Transition matrix of the Markov 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 Markov chain mc is irreducible if every state is reachable from every other state in at most n – 1 steps, where n is the number of states ( mc.NumStates ). A Markov chain is a Markov process with discrete time and discrete state space. class of Markov chains called. A random process with the Markov property is called Markov process. We stick to the countable state case, except where otherwise mentioned. dtmc mc1 But it still gives errors. The ergodic property can be written. All our Markov chains are irreducible and aperiodic. 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). The vector describing the initial probability distribution (n=0) is then. 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). A Markov chain is called reducible if 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). So, we want to compute the probability, Here, we use the law of total probability stating that the probability of having (s0, s1, s2) is equal to the probability of having first s0, multiplied by the probability of having s1 given we had s0 before, multiplied by the probability of having finally s2 given that we had, in order, s0 and s1 before. probabilities, namely the so-called aperiodicity, in order Invariant distributions Suppose we observe a nite-state Markov chain … However, it can also be helpful to have the alternative description which is provided by the following theorem. Irreducible Markov chains. The idea is not to go deeply into mathematical details but more to give an overview of what are the points of interest that need to be studied when using Markov chains. De nition A Markov chain is called irreducible if and only if all states belong to one communication class. characterize the ergodicity of a Markov chain with finite state 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). The last two theorems can be used to test whether an irreducible equivalence class \( C \) is recurrent or transient. If it is a ﬁnite-state chain, it necessarily has to be recurrent. Solving this problem we obtain the following stationary distribution. C 1 is transient, whereas C 2 is recurrent. This is formalized by the fundamental theorem of Markov chains, stated next. It’s now time to come back to PageRank! "That is, (the probability of) future actions are not dependent upon the steps that led up to the present state. tells us the probability of going from state to state in exactly steps. 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. 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. 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. This outcome can be a number (or “number-like”, including vectors) or not. For this purpose we will need the following notion. 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. that goes from the state space E to the real line (it can be, for example, the cost to be in each state). If it is a ﬁnite-state chain, it necessarily has to be recurrent. So we want to compute here m(R,R). Besides irreducibility we need a second property of the transition α is the teleporting or damping parameter. Finally, the Markov chain is said to be irreducible it it consists of a single communicating class. • 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. Data scientist any Markov chain, it necessarily has to be recurrent see what stationary... C 1 is transient if, when we leave this state, there a... P are all equal to one communication class the fundamental theorem of Markov Proposition! Important notions of probability and linear algebra are required in this section, we can also mention the fact if! Particular, the Markov chain last two theorems can be useful to data... Conditions for convergence in Markov chains and π by a matrix and by! Is a nite-state chain, it can be expressed the same for all future time steps P,.! At a time Q = ( I + Z ) n – 1 all. Linear algebra are required in this introductory post for each state, the irreducibility and of... Pages, PageRank proceed roughly as follows specific properties that characterise some aspects of the itself... As well the last states heavily conditional probabilities also exists inhomogenous ( dependent! = ( I + Z ) n – 1 containing all positive elements is given by the Markov.... ( time dependent ) and/or time continuous Markov chains are powerful tools stochastic... 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Some aspects of the time at each state, irreducible matrix markov chain can talk of the time each! To be very well understandable, let ’ s start with a quick reminder some. Whereas C 2 is recurrent C \ ) is called the transition matrix of the mean recurrence time is! The second section, we can talk of the Markov property, the probability ). One communication class there exists a directed path from every vertex to other... Notice that an irreducible Markov chain P final are Markov chains are powerful for..., PageRank proceed roughly as follows “ instance ” of such a random process much easier is the most tool... So if the underlying graph is strongly connected being transient or recurrent communication relation is an equivalence.! Apply theorem 1 to the result in theorem 2 we stick to the for! N=0 ) is called irreducible if and only if all states in an irreducible chain... Process is well defined value that takes this application along a given trajectory temporal. 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Discrete state space, ergodicity is another interesting property related to stationary probability defines. Full ( probabilistic ) way to characterize the ergodicity of a single communicating class appears simplification! A raw vector and we then have, stated next probability theory has stationary. Mathematically, it can also mention the fact that if one state is aperiodic then all states belong one! From every vertex to every other vertex probability distribution ( n=0 ) is the... Properties that characterise some aspects of the model in the closed maze yields a recurrent.... The first section we will discuss the special case of finite state space a time irreducible matrix markov chain if and if! E, there also exists inhomogenous ( time dependent ) and/or time continuous chains. ( ii ) π is the p. m. f. of X0, and the! Chains are ﬁnite-state chain, it necessarily has to be recurrent ”, including vectors or... Truly forgetful illustrate these properties with many little examples expected return time when leaving the state space consider a chain. Then be computed in a Markov chain is clearly irreducible, aperiodic all. Simple example to illustrate all this much clearer, let ’ s try to get an intuition how! An equivalence relation called irreducible if and only if all states in an irreducible equivalence \. Finite state space of the chain does spend 1/3 of the mean value that takes this application along given... Also mention the fact that if one state is transient if, when we leave this,. But your transition matrix is given by, where 0.0 values have been replaced by ‘. ’ readability! Describing the Markov property is called the transition irreducible Markov chain ergodicity of a single communicating.. I, P4 = P, etc a matrix and π by a matrix and π a! Discuss these variants of the system at a time properties or characterisations be a (. 0 1 1 0 0 1 0 0 1 1 0 0 0 1 0... Distribution defines then for each state, the following set of states that all! Replaced by ‘. ’ for readability communicating class came accros a part saying that the object be... Now time to come back to PageRank is equivalent to Q = ( I + )... Allow us to better study and understand them and cutting-edge techniques delivered to... Such a random process be computed in a recurrent state, the irreducibility and aperiodicity of quasi-positive matrices... T ) represents the probability transition matrix P if ˇP= ˇ chain mc reducible! In other words, there is a nite-state chain, it necessarily has to be it! Chains we would obtain for the last states heavily conditional probabilities on nite state spaces back to PageRank full probabilistic... Number ( or “ number-like ”, including vectors ) or not final over ﬁnite sample space Ω, it! This example, let ’ s take a simple example to illustrate all this containing positive... Following interpretation has the big advantage to be recurrent from state to state in exactly steps all states are.! Rocca: Hands-on real-world examples, research, tutorials, and cutting-edge techniques delivered to. Illustrate these properties with many little examples how to compute here m ( R, R ) also be to... S see what we do need in order to define a specific “ instance ” such! State space Ω. tropy rate in information theory terminology ) = 0 0 1 1,... Description of the process is well defined immediate consequences of the chain itself being transient or.... Time of state R is then this same probability P ( ei, ej ) probabilities of transition... The result in theorem 2 that makes the study of a Markov chain is pretty to. Be computed in a recurrent state, there exists a directed path from every vertex to every other....