Programs

# Introduction to Markov Chains: Prerequisites, Properties & Applications

Has it ever crossed your mind how expert meteorologists make a precise prediction of the weather or how Google ranks different web pages? How they make the fascinating python applications in real world. These calculations are complex and involve several variables that are dynamic and can be solved using probability estimates.

When Google introduced its PageRank algorithm, it revolutionized the web industry. And if you’re familiar with that algorithm, you must also know that it uses Markov chains. In our introduction to Markov chains, we’ll take a brief look at them and understand what they are. So, let’s get started.

## Pre-requisites

It’s essential to know a few concepts before we start discussing Markov chains. And most of them are from probability theory. Non-mathematically, you can define a random variable’s value as the result of a random event. So, for example, if the variable were the result of rolling a die, it would be a number whereas if it were a result of a coin flip, it would be a boolean (0 or 1). The set of these possible results could be continuous as well as discrete.

So we can say that a stochastic process is a collection of random variables that set indexes. That set represents different time instances. This set could be of real numbers (continuous process) or natural numbers (discrete process).

## Introduction to Markov Chains

Markov chains get their name from Andrey Markov, who had brought up this concept for the first time in 1906. Markov chains refer to stochastic processes that contain random variables, and those variables transition from a state to another according to probability rules and assumptions.

### What is the Markov Property?

There are plenty of groups of random processes, such as autoregressive models and Gaussian processes. Markov property makes the study of these random processes quite easier. A Markov property states that we wouldn’t get more information about the future outcomes of a process by increasing our knowledge about its past if we know its value at a particular time.

A more elaborate definition would be: Markov property says that the probability of a stochastic process only depends on its current state and time, and it is independent of the other states it had before. That’s why it’s a memoryless property as it only depends on the present state of the process.

A homogeneous discrete-time Markov chain is a Marko process that has discrete state space and time. We can say that a Markov chain is a discrete series of states, and it possesses the Markov property.

Here’s the mathematical representation of a Markov chain:

X = (Xn)nN=(X0, X1, X2, …)

## Properties of Markov Chains

Let’s take a look at the fundamental features of Markov chains to understand them better. We won’t delve too deep on this topic as the purpose of this article is to make you familiar with the general concept of Markov chains.

### Reducibility

Markov chains are irreducible. That means they have no reducibility if it can reach any state from another state. The chain doesn’t need to reach one state from another in just a single time step; it can do so in multiple time steps. If we can represent the chain with a graph, then the graph would be firmly connected.

### Aperiodic

Let’s say a state’s period is k. If k = 1, then this state is aperiodic when any kind of return to its state requires some multiple of k time-steps. When all states of a Markov chain are aperiodic, then we can say that the Markov chain is aperiodic.

### Transient and Recurrent States

When you leave a state, and there’s a probability that you can’t return to it, we say that the state is transient. On the other hand, if we can return to a state with probability 1, after we have left it, we can say that the property is recurrent.

There are two kinds of recurrent states we can have. The first one is the positive recurrent state that has a finite expected return time, and the second one is the null recurrent state that has an infinite expected return time. Expected return time refers to the mean recurrence time when we are leaving the state.

## Applications of Markov Chains

Markov chains find applications in many areas. Here are their prominent applications:

• Google’s PageRank algorithm treats the web like a Markov model. You can say that all the web pages are states, and the links between them are transitions possessing specific probabilities. In other words, we can say that no matter what you’re searching on Google, there’s a finite probability of you ending up on a particular web page.
• If you use Gmail, you must’ve noticed their Auto-fill feature. This feature automatically predicts your sentences to help you write emails quickly. Markov chains help in this sector considerably as they can provide predictions of this sort effectively.
• Have you heard of Reddit? It’s a significant social-media platform that’s filled with subreddits (a name for communities in Reddit) of specific topics. Reddit uses Markov chains and models to simulate subreddits for a better understanding of the same.

## Final Thoughts

It appears we have reached the end of our introduction to Markov chains. We hope you found this article useful. If you have any questions or queries, feel free to share them with us through the comments. We’d love to hear from you.

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## Is there any real life application of Markov Chains?

One of the most essential tests for dealing with separate trial procedures is the Markov chain. In finance and economics, Markov chains are used to represent a variety of events, such as market crashes and asset values. Markov chains are applied in a wide range of academic areas, including biology, economics, and even real-world scenarios. Parking lots have a set number of spots available, but how many are available at any one moment may be characterized using a Markov model based on a combination of numerous factors or variables. Markov chains are frequently used to create dummy texts, lengthy articles, and speeches.

## What does the term equilibrium mean with respect to Markov Chains?

The distribution πT is said to be an equilibrium distribution If πT P = πT. Equilibrium refers to a situation where the distribution of Xt does not change as we progress through the Markov chain. In fact, the distinguishing feature of a Markov chain is that the potential future states are fixed, regardless of how the process got to its current state. In other words, the likelihood of transitioning to any given condition is completely determined by the present state and the amount of time that has passed.

## Are Markov Chains time homogenous?

If the transition probability between two given state values at any two times relies only on the difference between those times, the process is time homogenous. There are conditions for a Markov chain to be homogeneous or non-homogeneous. The transition probabilities of a Markov chain are said to be homogenous if and only if they are independent of time. The Markov property is retained in non-homogeneous Markov chains (nhmc), although the transition probabilities may vary with time. This section lays forth the criteria that guarantee the presence of a variation limit in such chains, with the goal of applying them to simulated annealing. 