Markov Chains || Step-By-Step || ~xRay Pixy

Learn Markov Chains step-by-step using real-life examples.

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Video Chapters: Markov Chains

00:00 Introduction

00:19 Topics Covered

01:49 Markov Chains Applications

02:04 Markov Property

03:18 Example 1

03:54 States, State Space, Transition Probabilities

06:17 Transition Matrix

08:17 Example 02

09:17 Example 03

10:26 Example 04

12:25 Example 05

14:16 Example 06

16:49 Example 07

18:11 Example 08

24:56 Conclusion

In computer science, Markov problems are typically associated with Markov processes or Markov models. These are related to topics involving stochastic processes and probabilistic systems where future states depend only on the current state, not on the sequence of states that preceded it.

Artificial Intelligence (AI):

• Markov Decision Processes (MDP): Used in decision-making problems, especially in reinforcement learning.
• Hidden Markov Models (HMM): Widely used in speech recognition, handwriting recognition, and natural language processing.

Machine Learning:

• Probabilistic graphical models like HMM.
• Stochastic optimization techniques.

Data Science and Statistics:

• Statistical analysis using Markov Chains.
• Time series forecasting and data modeling.

Networking and Distributed Systems:

• Queueing theory and performance modeling using Markov chains.
• Reliability analysis.

Game Theory:

• Modeling strategies and decision-making under uncertainty.

Simulation and Modeling:

• Using Markov Chains to simulate random systems like traffic, communication networks, or biological processes.

Operations Research:

• Optimization problems using Markov Decision Processes.
• Applications in logistics and supply chain management.

Markov Chains are memoryless because of the Markov Property. In a Markov Chain, the future only depends on the current state, not on the entire history of how you got there.

Q. Why Does This Make Markov Chains Simpler to Analyze?

1. Less Data to Track: You only need to focus on the current state and the transition probabilities (how likely it is to move from one state to another). There’s no need to track or calculate the entire history.
2. Simplified Mathematics:
   Instead of dealing with complicated probabilities that depend on past steps, you can use simple equations based on the current state.
3. Efficient Predictions:
   For example, if you want to predict where the robot will be after 10 steps, you don’t have to trace every possible path it took before step 10. You just calculate step by step from the current state.

Key Components of the Markov Property:

Current State:

• The future depends only on the current state of the system, and not on how the system arrived at that state.

Transition Probability:

• The probability of moving from one state to another depends only on the current state, not on the sequence of past states.

Example: Weather Model

Imagine you are tracking the weather, which can either be Sunny (S) or Rainy (R) each day. The Markov Chain model assumes that the weather tomorrow depends only on today’s weather, not on any previous days.

Let’s say the probabilities are as follows:

• If it’s Sunny today, there is a 70% chance it will be Sunny tomorrow and a 30% chance it will be Rainy.
• If it’s Rainy today, there is a 40% chance it will be Sunny tomorrow and a 60% chance it will be Rainy.

Transition Matrix

We can represent this as a transition matrix where each element represents the probability of moving from one state (today’s weather) to another (tomorrow’s weather):

$$\left[\begin{array}{cc}P(\text{S}\to \text{S})& P(\text{S}\to \text{R})\\ P(\text{R}\to \text{S})& P(\text{R}\to \text{R})\end{array}\right]=\left[\begin{array}{cc}0.7& 0.3\\ 0.4& 0.6\end{array}\right]$$