Markov Chains Clearly Explained! Part - 1

Summary

In 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.