Understanding Markov Chains through Dice Rolls and Weather Models
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Chapter 1: Introduction to Markov Chains
Markov Chains represent a foundational concept in various domains, particularly in Machine Learning, where they underpin models like Hidden Markov Models (HMM) and Reinforcement Learning. Their applications extend to fields such as finance (for tracking stock price fluctuations) and engineering physics (like Brownian motion). A notable instance of Markov Chains in action is Google's PageRank algorithm, which was initially designed to rank web pages.
The allure of Markov Chains lies in their unique perspective on predictive modeling. Unlike traditional Machine Learning approaches that rely on historical data to forecast future events, Markov Chains focus solely on the current state to make predictions. This difference opens up intriguing possibilities for analysis.
Section 1.1: Predicting Weather with Markov Chains
Consider the challenge of forecasting the weather over the next 365 days for a particular city. A straightforward method might involve assigning probabilities for rainy and sunny days. For example, in London, where rain is common, we could set the probabilities as follows: P(Raining) = 1/3 and P(Sunny) = 2/3. By running a simple simulation in Python, we could generate potential weather outcomes.
However, this method has a significant limitation. In reality, the weather is not independent day-to-day; if it rains today, it’s more likely to rain tomorrow, and similarly for sunny days. To reflect this, we need to adjust our model so that if it rains today, the probability of rain tomorrow increases, and vice versa for sunny days.
To address this, we can utilize Markov Chains, which consist of states (rainy or sunny), events (transitioning from one state to another), and the probabilities associated with these events.
Section 1.2: Transition Probabilities in Markov Chains
In our weather example, we can represent the situation with a Markov Chain. Each state has probabilities for remaining in the same state or transitioning to another. This can be visualized as a transition matrix, which succinctly captures these probabilities.
For instance, the transition matrix for our weather model might indicate that the probability of moving from a sunny day to a rainy one is 0.3.
Chapter 2: Modeling Dice Rolls with Markov Chains
Next, let’s explore a more complex scenario using dice rolls. Imagine two players, A and B. Player A bets that two consecutive rolls will sum to 10, while Player B bets on rolling two consecutive sixes. The game continues until one player achieves their goal.
To analyze this, we again apply a Markov Chain. Player A’s winning combinations include rolling a 5 followed by another 5, a 4 and then a 6, or a 6 and then a 4. Player B only wins if they roll two sixes in succession.
The first video, "A Gentle Introduction to Markov Chain Monte Carlo by Prof. Dootika Vats," provides a clear overview of Markov Chains and their applications.
To structure our Markov Chain for this game, we define a series of states and their associated transition probabilities. The initial state comprises rolls that do not contribute to either player's win condition. The chain evolves based on the probabilities of each roll leading to a winning combination.
The second video, "Markov Chains Clearly Explained! Part - 1," elucidates the concept of Markov Chains and how they can be applied in various scenarios, including games and weather forecasting.
Final Thoughts: Exploring Markov Chains
This introduction aims to spark your interest in Markov Chains and their potential applications. I encourage you to experiment with different scenarios, such as varying the win conditions in our dice example or simulating different weather patterns. By doing so, you can deepen your understanding of this fascinating concept in probability and statistics. Feel free to share your thoughts and insights as you explore further!