Markov Chains Clearly Explained! Part - 1

Normalized Nerd

25 Oct 202009:24

Summary

TLDRIn this 'Normalized Nerd' video, the presenter introduces Markov chains, a concept used across various fields like statistics, biology, economics, physics, and machine learning. The video explains Markov chains through a restaurant example that serves three types of food with a predictable pattern based on the previous day's meal. The Markov property, which states that the future state depends only on the current state, is highlighted. The presenter also discusses the stationary distribution, or equilibrium state, which represents the long-term probability distribution of states. The video demonstrates how to find this distribution through simulation and linear algebra, using an adjacency matrix and eigenvector equations to solve for the stationary state.

Takeaways

😀 Markov chains are used in various fields such as statistics, biology, economics, physics, and machine learning.
🍔 The video uses a restaurant example with three food types (hamburger, pizza, hot dog) to explain Markov chains, where the probability of serving a food type depends on what was served the previous day.
🔢 A Markov chain is represented by a diagram with weighted arrows indicating the probability of transitioning from one state to another.
📈 The Markov property states that the future state depends only on the current state, not on the sequence of previous states.
🎯 The sum of the probabilities (weights) of outgoing transitions from any state must equal 1, as they represent a complete probability distribution.
🚶‍♂️ A random walk along the chain can be used to explore the probabilities of each state over time.
📊 After many steps, the probabilities converge to a stationary distribution, also known as the equilibrium state, which does not change over time.
📚 Linear algebra can be used to find the stationary state more efficiently than simulation, by representing the Markov chain as an adjacency matrix and solving for the eigenvector corresponding to the eigenvalue of 1.
🔍 The stationary state can reveal long-term probabilities, such as the restaurant serving hamburgers about 35% of the time, pizza 21%, and hot dogs 44% of the time.
🔬 The eigenvector method also allows for the detection of multiple stationary states by checking for multiple eigenvectors with an eigenvalue of 1.
📈 The results from the linear algebra method can be compared with simulation results to validate the findings and understanding of the Markov chain.

Q & A

What is a Markov chain?
-A Markov chain is a mathematical concept used in various fields such as statistics, biology, economics, physics, and machine learning. It represents a sequence of possible events where the probability of each event depends only on the state attained in the previous event.
Why are Markov chains interesting?
-Markov chains are interesting because they simplify the process of predicting future states by only considering the current state, without needing to know the sequence of events that led up to it. This makes them useful for modeling real-world problems with complex dependencies.
What is the restaurant example in the script illustrating?
-The restaurant example illustrates how a Markov chain works. The restaurant serves only one of three foods each day, and the probability of serving a particular food depends on what was served the previous day. This creates a predictable pattern that can be represented as a Markov chain.
What does the term 'state' refer to in the context of Markov chains?
-In the context of Markov chains, 'state' refers to a situation or condition that can result in a particular outcome. In the restaurant example, the state is the type of food being served on a given day.
What is the Markov property?
-The Markov property is the fundamental characteristic of a Markov chain, stating that the probability of transitioning to a future state depends only on the current state and not on the sequence of events that preceded it.
Why is the sum of the weights of outgoing arrows from any state equal to 1 in a Markov chain?
-The sum of the weights of the outgoing arrows from any state equals 1 because these weights represent probabilities. For probabilities to be valid, they must add up to one, ensuring that all possible outcomes are accounted for.
What is a stationary distribution in the context of Markov chains?
-A stationary distribution, also known as the equilibrium state, is a probability distribution of the states in a Markov chain that does not change over time. It represents the long-term probabilities of being in each state after a large number of transitions.
How can you find the stationary state of a Markov chain?
-One way to find the stationary state of a Markov chain is through simulation, running a large number of steps and observing the convergence of probabilities. Alternatively, linear algebra can be used by setting up and solving the eigenvector equation for the transition matrix with the eigenvalue equal to 1.
What is an adjacency matrix and how is it used in representing a Markov chain?
-An adjacency matrix is a way to represent a directed graph, such as a Markov chain, where the elements of the matrix denote the weight of the edge connecting two vertices. In the context of Markov chains, it is called the transition matrix and is used to calculate the probabilities of transitioning from one state to another.
How does the script demonstrate the use of linear algebra to find the stationary state?
-The script demonstrates the use of linear algebra by showing how to set up an eigenvector equation for the transition matrix with an eigenvalue of 1. Solving this equation yields the stationary state, which represents the long-term probability distribution of the states in the Markov chain.
How can you determine if there are multiple stationary states in a Markov chain?
-To determine if there are multiple stationary states, you can look for the existence of more than one eigenvector with an eigenvalue equal to 1 for the transition matrix. Each eigenvector represents a different stationary state.

Outlines

00:00

🍔 Introduction to Markov Chains with a Restaurant Example

The video begins with an introduction to Markov chains, a concept used across various disciplines like statistics, biology, economics, physics, and machine learning. The presenter uses a restaurant example to illustrate the concept, where the restaurant serves only one of three foods—hamburger, pizza, or hot dog—based on what was served the previous day. This creates a predictable pattern, represented by a Markov chain diagram with weighted arrows indicating the transition probabilities between states. The Markov property is highlighted, which states that the future state depends solely on the current state, not on the sequence of past states. This property simplifies the handling of complex real-world problems.

05:01

📊 Markov Chain Properties and Simulation of a Random Walk

The video continues by discussing the properties of Markov chains. It emphasizes that the sum of the transition probabilities from any state must equal 1, as they represent a complete set of probabilities. The presenter then introduces a simulation of a random walk along the Markov chain to explore the probability distribution of states over time. By conducting a random walk for 10 consecutive days, the presenter demonstrates how the probabilities for each food item change and eventually seeks to determine if these probabilities converge to fixed values in the long run, indicating a stationary or equilibrium state.

🔍 Efficient Calculation of Stationary States Using Linear Algebra

The presenter acknowledges that the simulation method for finding stationary states may not be efficient and suggests using linear algebra as a more effective approach. The Markov chain is represented by an adjacency matrix, also known as the transition matrix, where each element denotes the transition probability between states. The goal is to find the stationary state, which is a row vector representing the long-term probability distribution of states. The presenter explains that this can be achieved by solving for the eigenvector of the transition matrix with an eigenvalue of 1, which corresponds to the stationary distribution. The method is shown to yield a close result to the one obtained from the simulation, validating the stationary state probabilities for the restaurant example.

Mindmap

Keywords

💡Markov Chains

Markov Chains are stochastic models that describe a sequence of possible events where the probability of each event depends only on the state attained in the previous event. In the video, Markov Chains are introduced as a concept used across various fields like statistics, biology, economics, physics, and machine learning. The script uses a restaurant example to illustrate how the probability of serving a particular food item depends solely on what was served the previous day, showcasing the Markov property.

💡Normalized Nerd

Normalized Nerd appears to be the name of the channel or the host presenting the video. It is not a technical term but rather a branding element that adds personality and identity to the content. The channel likely focuses on explaining complex topics in a more accessible and 'normalized' manner, as suggested by the name.

💡Machine Learning

Machine Learning is a subset of artificial intelligence that enables computers to learn from data and improve at tasks without being explicitly programmed. In the context of the video, machine learning is mentioned as one of the fields where Markov Chains are applied, indicating the video's relevance to both theoretical concepts and practical applications in technology.

💡State

In the context of Markov Chains, a 'state' represents a situation or condition that can be occupied at a certain time step. The video uses the term 'state' to describe the different types of food served by the restaurant on any given day, with each food item representing a unique state in the Markov Chain.

💡Transition

A 'transition' in Markov Chains refers to the movement from one state to another. The video script describes transitions with weighted arrows, where each arrow represents the probability of moving from the current state (e.g., serving hamburgers) to a future state (e.g., serving pizza). Transitions are fundamental to understanding how Markov Chains model sequences of events.

💡Markov Property

The Markov Property is the core characteristic of a Markov Chain, stating that the future state depends only on the current state and not on the sequence of events that preceded it. The video emphasizes this property by explaining that knowing what the restaurant served on the current day is sufficient to predict what they will serve the next day, without needing to know the entire history of served items.

💡Stationary Distribution

The 'Stationary Distribution' or 'Equilibrium State' in a Markov Chain refers to the probability distribution of the states that does not change over time. The video discusses how, after simulating a random walk for a large number of steps, the probabilities of serving each food item converge to fixed values, indicating the stationary distribution of the Markov Chain.

💡Adjacency Matrix

An 'Adjacency Matrix' is a way to represent a directed graph, where the elements of the matrix denote the weight of the edge connecting two vertices. In the video, the Markov Chain is represented by an adjacency matrix, which is also called the transition matrix, to efficiently model the transitions between states.

💡Eigenvector

An 'Eigenvector' is a non-zero vector that does not change its direction after a linear transformation represented by a matrix. In the context of the video, the stationary state of the Markov Chain is found by solving for the eigenvector of the transition matrix where the eigenvalue is 1, which helps in determining the long-term probabilities of each state.

💡Linear Algebra

Linear Algebra is a branch of mathematics that deals with vector spaces and linear equations. The video suggests using linear algebra as a more efficient method to find the stationary state of a Markov Chain, by solving the eigenvector equation related to the transition matrix.

Highlights

Introduction to Markov chains and their applications in various fields such as statistics, biology, economics, physics, and machine learning.

Explanation of the concept of Markov chains with an example of a restaurant serving three types of food with a specific rule based on the previous day's serving.

Use of weighted arrows in a diagram to represent the transitions between states in a Markov chain.

The Markov property, which states that the future state depends only on the current state, not on the sequence of previous states.

The requirement that the sum of the weights of outgoing arrows from any state in a Markov chain must equal 1, representing the total probability.

Discussion of special properties of certain Markov chains that deserve individual videos.

Exploring the Markov chain through a random walk simulation to understand the probability distribution of states over time.

The observation that probabilities may converge to fixed values in the long run, known as the stationary distribution or equilibrium state.

A Python script used to simulate a random walk for 100,000 steps to observe the convergence of probabilities to the stationary distribution.

Introduction to a more efficient method to find the stationary state using linear algebra and adjacency matrices.

Representation of a Markov chain as a directed graph and its translation into an adjacency matrix for efficient analysis.

The use of a row vector to represent the probability distribution of states and its multiplication with the transition matrix to find future probabilities.

The concept of a stationary state as an eigenvector equation where the eigenvalue is 1, representing a state that does not change over time.

The condition that the elements of the stationary state vector must sum up to 1, as it represents a probability distribution.

The method to determine if there are multiple stationary states by checking for more than one eigenvector with an eigenvalue of 1.

Comparison of the stationary state found using linear algebra with the results from the simulation, validating the findings.

Encouragement for viewers to request more videos on Markov chains and a call to action to subscribe and share the video.