Making Sense of Uncertainty using Markov Chains

2024 Nov 25th

Imagine you need to plan a weekend trip to a nearby city, but the weather forecast is uncertain. You have to decide whether to go or not, but the uncertainty makes it challenging. This is where Markov Chains can help! But first, let's understand what probability is.

What is Probability?

Probability is a measure of how likely an event is to occur. It is expressed as a number between 0 and 1 (or 0% and 100%). A probability of 0 means the event is impossible, while a probability of 1 means the event is certain. For example, the probability of rolling a 6 on a fair six-sided die is 1/6 or approximately 0.167 (16.7%).

Understanding Markov Chains

A Markov Chain is a mathematical model used to make decisions when the future is uncertain. It describes the probability of transitioning between different states, like sunny, rainy, or cloudy weather.

Here's how it works:
1. Identify the States : In our weather example, the states are sunny, rainy, and cloudy.
2. Determine the Transition Probabilities : Using historical data, we find the probability of changing from one state to another. For example, if it is sunny today, there might be a 70% chance of it being sunny tomorrow, a 20% chance of it being cloudy, and a 10% chance of it being rainy.
3. Make a Decision : Based on these probabilities, you decide whether to go on your trip or not. If the chance of rain is too high, you might want to postpone your trip.

Visualizing a Markov Chain

Let's visualize a simple Markov Chain for our weather example: This table shows the transition probabilities between different weather states. (Note: The data table is a fictional example of weather probabilities for an area, written as "state1 today" -> "state2 tomorrow" : probability of state 2.)

```
Sunny -> Sunny: 0.7
Sunny -> Cloudy: 0.2
Sunny -> Rainy: 0.1

Cloudy -> Sunny: 0.3
Cloudy -> Cloudy: 0.4
Cloudy -> Rainy: 0.3

Rainy -> Sunny: 0.2
Rainy -> Cloudy: 0.3
Rainy -> Rainy: 0.5
```

For instance, if it is sunny today, there is a 70% chance it will be sunny tomorrow, a 20% chance it will be cloudy, and a 10% chance it will be rainy. You can then make a decision based on this knowledge of probabilities.

Advantages of Markov Chains

1. Simplicity : Markov Chains are relatively simple to understand and implement, making them accessible for a wide range of applications.
2. Versatility : They can be applied to various fields, from finance to epidemiology, making them a versatile tool for decision-making.
3. Predictive Power : Markov Chains provide a structured way to predict future states based on current information, helping to reduce uncertainty.

Disadvantages of Markov Chains
1. Memorylessness : Markov Chains assume that the future state depends only on the current state, not on the sequence of events that preceded it. This assumption may not always hold true in real-world scenarios.
2. Data Requirements : Accurate transition probabilities require substantial historical data, which may not always be available.
3. Complexity in Large Systems : For systems with many states (or possible outcomes), the Markov Chain can become complex and computationally intensive.

Alternative Mathematical Processes

While Markov Chains are powerful, there are other mathematical techniques that can be used for decision-making under uncertainty:
1. Bayesian Networks : These are graphical models that represent the probabilistic relationships among a set of variables. They are particularly useful for situations where the relationships between variables are complex and interdependent.
2. Hidden Markov Models (HMMs) : These are extensions of Markov Chains where the system being modeled is assumed to be a Markov process with hidden states. HMMs are useful in scenarios where the states are not directly observable.
3. Monte Carlo Simulations : These are computational algorithms that rely on repeated random sampling to obtain numerical results. They are useful for modeling systems with a high degree of uncertainty and complexity.

Applications of Markov Chains

Markov Chains are used in many high-stakes decision-making situations, such as:

Predicting Stock Market Trends : Financial analysts use Markov Chains to model the probability of different market conditions (bullish, bearish, neutral) and make informed investment decisions.
Analyzing the Spread of Diseases : Epidemiologists use Markov Chains to predict the spread of infectious diseases like COVID-19, helping inform public health policies and interventions.
Modeling Voter Behavior : Political scientists use Markov Chains to analyze voter behavior and predict election outcomes, aiding in campaign strategies and policy-making.