Understanding Markov Chains: Key Properties, Applications and Advantages

Markov chains are mathematical systems that transition between states, where the probability of each state depends only on the previous one. Understanding what Markov chains is easier with real-world examples that show how they model random processes.

In this blog, you’ll explore the applications of Markov chains across various fields and show you how these systems are used to model random processes. 

What is Markov Chains? Understanding Definitions and Various Representation Methods 

A Markov chain is a stochastic process that transitions between a finite or countable set of states.

Markov chains include variations like absorbing Markov chains and hidden Markov models, used in diverse applications.

The Markov property makes Markov chains unique, which means that the system's future state depends only on the present state, not on the sequence of events that preceded it. This is known as the "memoryless" nature of Markov chains.

Memoryless Property (Markov Property): The next state depends only on the current state, not on the history of states.

Ways to Represent Markov Chains:

Markov chains can be represented in different ways to analyze system dynamics.

• State Transition:

The simplest representation is through a state transition diagram. In this diagram, states are represented as nodes, and transitions are shown as directed edges between the nodes. 

Each edge is labeled with a probability, representing the likelihood of transitioning from one state to another.

Example: 

Imagine a weather model with two states—“Sunny” and “Rainy.” 

The state transition diagram would show arrows from "Sunny" to "Rainy" and vice versa, each with a corresponding probability (e.g., the probability of going from "Sunny" to "Rainy" might be 0.3).

• Transition Matrix:

A transition matrix is a square matrix used to describe the transitions of a Markov chain. Each entry in the matrix represents the probability of moving from one state to another.

Example:

This matrix tells us that there’s a 70% chance of staying sunny, and a 30% chance of transitioning to rainy weather from sunny weather. Similarly, from rainy weather, there's a 60% chance of staying rainy and a 40% chance of switching to sunny.

• Probability Distribution:

A probability distribution represents the likelihood of each state in a Markov chain at any given time. It’s often used to show the distribution of states after a certain number of steps, and it can be represented as a vector. Each element in the vector gives the probability of being in each state.

Example: If after one step, the probability distribution for the weather model is [0.6,0.4] it means there’s a 60% chance of sunny and a 40% chance of rainy.

Also Read: Types of Probability Distribution [Explained with Examples]

Understanding Higher-Order Markov Chains and Techniques for Estimation

Markov chains are often introduced as first-order models, but higher-order Markov chains are needed for complex patterns like language processing and financial modeling.

These chains consider multiple previous states, not just the immediate last one. This is useful when the memory of past states influences the current state.

High-order Markov chains consider multiple past states to predict future outcomes. They are used in language models for text generation, economic forecasts to analyze market trends, and multi-step decision-making in AI and robotics.

Techniques for Estimation:

• Maximum Likelihood Estimation (MLE):

MLE is used to estimate the parameters of a Markov chain, such as transition probabilities, from observed data. The idea is to find the set of transition probabilities that maximize the likelihood of the observed transitions.
Example: In our weather model, we would collect data on how often "Sunny" transitions to "Rainy" and vice versa, and use MLE to estimate the transition probabilities from this data.

• Smoothing Techniques:

Smoothing techniques are used to improve the estimates of transition probabilities, especially when there is sparse or incomplete data. These techniques adjust the probabilities to avoid assigning zero probability to unseen transitions.
Example: In the weather model, if the observed data doesn’t show any direct transition from "Sunny" to "Rainy," smoothing techniques would allow for a small non-zero probability, even if the transition wasn’t observed in the data.

Also Read: Top 12 Spark Optimization Techniques: Boosting Performance and Driving Efficiency

Now that we've covered the basics, let's dive into the different types of Markov chains and their key properties.

Types and Key Properties of Markov Chains

Markov chains can be categorized based on their structure and behavior. 

Markov chains can be broadly classified based on how time is treated and how transitions between states occur.  

• Discrete-Time Markov Chains:

In discrete-time Markov chains, the system transitions from one state to another at fixed, regular time intervals. These are the most common type of Markov chains, where time is divided into discrete steps, and the system makes a transition at each step.

Example: Predicting whether it will be sunny or rainy on each day based on the previous day's weather, with transitions happening at the end of each day.

• Continuous-Time Markov Chains:

Transitions can happen at any point in continuous-time Markov chains, and an exponential distribution typically models the time between transitions. 

These are used when events occur randomly over time rather than at fixed intervals.

Example: Modeling the time until the next event occurs, such as the time between customer arrivals at a service desk or the time between system failures in an industrial system.

Also Read: Hidden Markov Model in Machine Learning and Its Applications

Now that we've covered the types of Markov chains, let's delve into the key properties that define their behavior and stability over time.

Key Properties of Markov Chains

Let’s explore these key properties in more detail.

• Reducibility:

A Markov chain is reducible if it can get from any state to any other state in a finite number of steps. In other words, no state is isolated, and there are pathways connecting all states. If a Markov chain is reducible, it can eventually transition from one state to any other.

Example: In a weather model with states "Sunny," "Rainy," and "Cloudy," the chain is reducible if you can transition from any state to any other state, either directly or through intermediate states.

• Aperiodicity:

A Markov chain is aperiodic if it does not get trapped in a cycle and can return to a state at irregular intervals. In contrast, a periodic Markov chain has a fixed period where transitions happen at regular intervals, and it cannot return to a state at irregular times.

Example: In the weather model, if the chain can return to the "Sunny" state after 1 day, 3 days, or 5 days, it is aperiodic, meaning the system doesn’t follow a fixed cycle.

• Transient and Recurrent States:
  • Transient states: These are states that, once left, may never be visited again. If the system moves into a transient state, it might never return to it.
  • Recurrent states: These states are ones that, once visited, will eventually be revisited. The chain is guaranteed to return to these states.

Example: In a customer service model, a transient state could be a "busy signal" where customers might never call again. A recurrent state could be "waiting in the queue," where customers will eventually get through, regardless of how many times they enter the queue.

• Ergodicity:

A Markov chain is ergodic if it is both irreducible (can get from any state to any other) and aperiodic (no regular cycles). Ergodicity is a critical property because it ensures that the system will eventually reach a stable, long-term behavior that does not depend on the initial state.

Example: In a simple random walk model, where each state is reachable from any other and transitions happen at irregular intervals, the chain would be ergodic if, over time, it reaches a stationary distribution.

Now that we've covered the core properties, let’s look at how Markov chains are put to work across different industries, making a real impact.

Key Applications of Markov Chains in Various Fields

Below is a table that highlights key areas where Markov chains play an essential role.

Markov chains play a crucial role in various industries by modeling probabilistic transitions. In finance, they help predict market trends and stock price movements, with tools like Black-Scholes models and algorithmic trading strategies relying on them. 

Virtual assistants like Siri and Alexa use Hidden Markov Models (HMMs) for speech recognition, enabling accurate voice commands. 

In e-commerce, companies like Amazon and Shopify leverage Markov chains to model customer churn, optimizing retention strategies and personalized marketing.

Also Read: 5 Breakthrough Applications of Machine Learning

Now, let’s weigh the benefits and potential drawbacks of using this powerful tool.

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Advantages and Limitations of Markov Chains

Markov chains offer numerous advantages when modeling real-world systems, but they also come with challenges that need to be addressed.  

Advantages of Markov Chains

Markov chains are popular for their simplicity and effectiveness in modeling systems with probabilistic transitions. They offer many advantages in practical applications:

• Simplification of Complex Systems:

Markov chains simplify complex systems by breaking them down into states with defined transition probabilities, making analysis more structured. This approach is widely used in fields like economics, AI, and biology, where predicting future states based on current conditions helps in decision-making and optimization.

• Flexibility in Modeling:

Whether you're dealing with discrete or continuous processes, Markov chains offer flexibility to adapt to various problem domains like finance, healthcare, or social media behavior.

• Ease of Implementation:

Markov chains are relatively simple to implement, especially when compared to more complicated models like neural networks. This ease of implementation makes them accessible for both beginners and experts.

Also Read: 16 Best Neural Network Project Ideas & Topics for Beginners [2025]

While Markov chains offer valuable insights, they also have limitations, such as assuming future states depend only on the present, which may not always hold in complex systems.

Limitations of Markov Chains

These challenges often require additional strategies to overcome:

• Increased Dimensionality:

As the number of states increases, the complexity of the transition matrix grows exponentially, making computations more difficult and memory-intensive.

• Data Sparsity:

Markov chains rely on having enough data to estimate transition probabilities accurately. However, estimating these probabilities can become unreliable if the data is sparse.

• Memory Requirements:

As the state space becomes larger, storing and computing the transition probabilities for all possible states becomes more memory-intensive.

• Model Overfitting:

Markov chains can sometimes overfit the data, especially in high-dimensional systems, where the model starts to capture noise instead of generalizable patterns.

Mastering Markov chains is key in data science, and upGrad offers structured learning to apply them effectively.

How Can upGrad Support Your Growth in Data Science?

upGrad’s curriculum is designed to provide you with a comprehensive understanding of Markov chains and their applications in data science. upGrad provides hands-on training in stochastic modeling, covering real-world applications and industry-relevant tools.

Check out some of the top courses to help get you started:

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