Markov Chains Explained: How Systems Predict What Happens Next
Markov Chains Explained: How Systems Predict What Happens Next
Introduction
Many real-world systems are not static. They continuously move from one condition to another. A customer visits a website, views a product, adds it to a cart, and eventually either purchases the product or leaves. A server can move from a healthy state to a degraded state and later recover. Weather can change from sunny to cloudy and then to rainy. A financial account can move between different risk categories over time.
When we have this type of sequential behavior, one interesting question naturally appears:
Can we predict what is likely to happen next by looking at the current state of the system?
This is where Markov Chains become useful.
A Markov Chain is a mathematical model used to represent a system that moves between a set of possible states, where the probability of the next state depends primarily on the current state. The model is relatively simple, but the underlying idea is extremely useful in probability, statistics, data science, artificial intelligence, recommendation systems, reliability engineering, web analytics, and sequential data analysis.
For data engineers, Markov Chains are particularly interesting because modern applications continuously generate event sequences. Kafka streams, application logs, customer journeys, machine telemetry, transaction histories, and website events can all be transformed into state-transition data and analyzed using probabilistic models.
What Is a Markov Chain?
A Markov Chain is a stochastic model that describes how a system moves from one state to another over a sequence of time steps.
The most important concept is the Markov Property.
In simple terms:
The probability of what happens next depends on the current state rather than the complete history of previous states.
Consider a simple weather model with three possible states:
Suppose the weather is currently Sunny. We might estimate that tomorrow there is a 70% probability of remaining Sunny, a 25% probability of becoming Cloudy, and a 5% probability of becoming Rainy.
The model therefore uses the current state—Sunny—to estimate the next state.
It does not necessarily need to remember whether the previous two days were Rainy, Cloudy, or Sunny.
That assumption is what makes the model a Markov model.
Understanding the Markov Property
Suppose a system has passed through the following sequence:
Monday → Tuesday → Wednesday → Thursday
If we want to predict Thursday's state, a Markov model assumes that the current state on Wednesday contains the relevant information needed to determine the probability distribution for Thursday.
Mathematically, the Markov property can be written as:
P(Xt+1∣Xt,Xt−1,...,X0)=P(Xt+1∣Xt)P(X_{t+1}|X_t,X_{t-1},...,X_0) = P(X_{t+1}|X_t)
This looks complicated, but the idea is straightforward.
The left side says:
What is the probability of the next state given the entire history?
The right side says:
What is the probability of the next state given only the current state?
For a Markov process, these are treated as equivalent.
This does not mean that history is completely irrelevant in the real world. Instead, it means that the model assumes the current state contains enough information about the past for predicting the next state.
That distinction is important.
A Simple Real-World Example
Consider an online shopping application.
A user might move through several states:
Home → Search → Product → Cart → Purchase
But the user does not always follow the same path.
A user currently viewing a product might:
Add the product to the cart.
Return to search.
View another product.
Leave the website.
We can represent these possibilities using probabilities.
For example:
If a user is currently in the Product state, the model can estimate the probability of each possible next action.
This is a simple example of a Markov Chain.
States: The Building Blocks of a Markov Chain
A state represents the current condition of a system.
The definition of a state depends entirely on the problem being modeled.
For a customer analytics system, states might be:
New Visitor, Registered User, Active User, Premium User, Churned User
For a server monitoring system:
Healthy, Warning, Degraded, Failed
For a financial system:
Low Risk, Medium Risk, High Risk, Default
For an e-commerce application:
Home, Search, Product, Cart, Checkout, Purchase
The important thing is that the states should represent meaningful conditions that allow us to describe how the system changes over time.
Transition Probability
A transition means moving from one state to another.
The probability associated with that movement is called the transition probability.
Suppose a server has two states:
Healthy and Failed.
We might observe the following behavior:
This tells us that a healthy server has a 95% probability of remaining healthy during the next observation period and a 5% probability of failing.
Similarly, a failed server has a 30% probability of recovering and a 70% probability of remaining failed.
The Transition Matrix
When a Markov Chain contains multiple states, representing every transition individually can become difficult.
A more convenient representation is a transition matrix.
Consider a weather system with three states:
Sunny, Cloudy, Rainy
We could define the transition matrix as:
Each row represents the current state, while each column represents the possible next state.
For example, the first row means that when the current state is Sunny:
Probability of Sunny tomorrow = 0.70
Probability of Cloudy tomorrow = 0.25
Probability of Rainy tomorrow = 0.05
The probabilities in every row must add up to 1 because the system must transition into one of the possible states.
For Sunny:
0.70+0.25+0.05=10.70 + 0.25 + 0.05 = 1
The same principle applies to every other row.
Visual Representation of a Markov Chain
The transition matrix is useful for calculations, but a graph often makes the concept easier to understand.
0.25
┌────────────────┐
│ ▼
┌────────┐ ┌─────────┐
│ Sunny │ │ Cloudy │
└────────┘ └─────────┘
│ ▲ │ │
.05│ │.70 .30│ │.30
│ │ │ │
▼ │ ▼ │
┌─────────────────────────┐
│ Rainy │
└─────────────────────────┘
Each node represents a state and each arrow represents a possible transition.
The number attached to an arrow represents the probability of that transition.
Initial State Distribution
A Markov Chain also needs an initial probability distribution.
Suppose we start with:
This can be represented as:
π0=[0.60, 0.30, 0.10]\pi_0 = [0.60,\ 0.30,\ 0.10]
This tells us the probability distribution of the system at the beginning of our analysis.
Once we have the initial distribution and transition matrix, we can calculate the probability distribution at future time steps.
Predicting Future States
The next state distribution can be calculated using matrix multiplication:
π1=π0P\pi_1 = \pi_0P
After another transition:
π2=π1P\pi_2 = \pi_1P
And after several transitions:
πn=π0Pn\pi_n = \pi_0P^n
This is one of the most important mathematical relationships in Markov Chain analysis.
Instead of manually calculating every possible sequence, matrix multiplication allows us to efficiently calculate the probability distribution after multiple transitions.
Markov Chains with Python
We can implement the previous weather example using Python and NumPy.
import numpy as np
# Transition matrix
P = np.array([
[0.70, 0.25, 0.05],
[0.30, 0.40, 0.30],
[0.20, 0.35, 0.45]
])
# Initial probability distribution
initial = np.array([0.60, 0.30, 0.10])
# Calculate probability distribution after one transition
next_state = initial @ P
print(next_state)
The expression:
initial @ P
performs matrix multiplication.
The resulting vector represents the probability distribution of Sunny, Cloudy, and Rainy after one transition.
We can continue the calculation:
state = initial
for day in range(10):
state = state @ P
print(f"Day {day + 1}: {state}")
This allows us to observe how the probability distribution changes over time.
Simulating an Individual Markov Chain
Another approach is to simulate one possible sequence instead of calculating the complete probability distribution.
import random
states = ["Sunny", "Cloudy", "Rainy"]
transitions = {
"Sunny": [0.70, 0.25, 0.05],
"Cloudy": [0.30, 0.40, 0.30],
"Rainy": [0.20, 0.35, 0.45]
}
current_state = "Sunny"
for _ in range(15):
print(current_state)
current_state = random.choices(
states,
weights=transitions[current_state]
)[0]
A simulation might produce a sequence such as:
Sunny → Sunny → Cloudy → Rainy → Rainy → Cloudy → Sunny → Cloudy
Another execution could produce a completely different sequence.
That is expected because the model is probabilistic.
The transition probabilities remain the same, but the individual outcome is random.
Markov Chains and Long-Term Behavior
One of the most interesting aspects of Markov Chains is their ability to model long-term behavior.
Imagine repeatedly applying the transition matrix:
π0P\pi_0P
then:
π0P2\pi_0P^2
then:
π0P3\pi_0P^3
and so on.
For many suitable Markov Chains, the probability distribution eventually approaches a stable distribution.
This is called the stationary distribution.
The stationary distribution satisfies:
π=πP\pi = \pi P
In other words, if the system has reached its stationary distribution, applying the transition matrix again does not change the distribution.
This is useful when we are interested in long-term system behavior rather than a single future prediction.
Example: Customer Lifecycle
Suppose an application has the following customer states:
Customers can move between these states.
For example:
New → Active → Premium
But some customers may move:
Active → Inactive
and eventually:
Inactive → Churned
Historical customer data can be used to estimate these transition probabilities.
Once the transition matrix is built, we can estimate questions such as:
What percentage of customers may become premium users?
How likely is an active customer to become inactive?
What percentage of customers may eventually churn?
What does the long-term customer distribution look like?
This is where Markov Chains become particularly interesting for business analytics.
Markov Chains in Data Engineering
Markov Chains are especially relevant when working with event-driven systems.
Modern applications generate huge volumes of sequential events.
For example:
Application
↓
Kafka
↓
Stream Processing
↓
S3 / Data Lake
↓
Spark / Glue
↓
Redshift / Snowflake
↓
Analytics
↓
Markov Model
Imagine an e-commerce platform producing events such as:
login
search
product_view
add_to_cart
checkout
purchase
logout
These events can be stored in a data lake or data warehouse.
A data engineering pipeline can transform the raw events into state transitions.
For example:
After aggregating millions of such transitions, we can calculate transition probabilities.
For example:
P(Purchase∣AddToCart)=Purchase TransitionsTotal AddToCart TransitionsP(Purchase|AddToCart) = \frac{Purchase\ Transitions}{Total\ AddToCart\ Transitions}
This converts raw event data into a probabilistic model of user behavior.
A Practical Architecture for Event-Based Markov Analysis
A realistic data platform might look like this:
┌──────────────────┐
│ Web / Mobile │
│ Application │
└────────┬─────────┘
│
▼
┌──────────────────┐
│ Kafka │
│ Event Stream │
└────────┬─────────┘
│
┌───────────┴───────────┐
▼ ▼
┌────────────────┐ ┌────────────────┐
│ Stream │ │ Raw Data │
│ Processing │ │ Lake / S3 │
└───────┬────────┘ └───────┬────────┘
│ │
└───────────┬───────────┘
▼
┌──────────────────┐
│ Spark / Glue / │
│ ETL Processing │
└────────┬─────────┘
│
▼
┌──────────────────┐
│ Data Warehouse │
│ Redshift/Snowflake│
└────────┬─────────┘
│
▼
┌──────────────────┐
│ State Transition │
│ Aggregation │
└────────┬─────────┘
│
▼
┌──────────────────┐
│ Markov Chain │
│ Probability Model │
└────────┬─────────┘
│
▼
┌──────────────────┐
│ Prediction / │
│ Business Insights│
└──────────────────┘
The important point is that the Markov Chain does not replace the data platform.
Instead, the data platform provides the historical event data required to construct the model.
Markov Chains in Recommendation Systems
Recommendation systems often need to understand what users are likely to do next.
Suppose a user has just viewed a laptop.
Historical data might show:
The recommendation engine can use this information as one input into a larger recommendation system.
For example, if users frequently move from one product category to another, the system can identify those behavioral patterns.
Markov Chains therefore provide an interpretable way of modeling sequential user behavior.
Markov Chains in System Reliability
Markov Chains can also be used to model the lifecycle of infrastructure components.
Consider a cloud service with four states:
Healthy → Warning → Degraded → Failed
The service can also recover:
Degraded → Healthy
and:
Failed → Degraded
Historical monitoring data can be used to estimate how frequently these transitions occur.
A reliability model can then help answer questions such as:
What is the probability of reaching a failed state?
How frequently does the system enter a degraded state?
How long does the system remain in each state?
What does the long-term operational state distribution look like?
This makes Markov Chains useful alongside monitoring, observability, and reliability engineering.
Absorbing Markov Chains
Some systems contain states from which the system cannot return.
Consider a simplified customer lifecycle:
Active → Churned
If Churned is considered permanent, then:
P(Churned→Churned)=1P(Churned \rightarrow Churned)=1
Such a state is called an absorbing state.
An absorbing Markov Chain can therefore be useful when a process has terminal states.
Examples include:
Customer permanently churning
Loan reaching a final state
Application reaching a terminal workflow state
Machine being permanently decommissioned
Process completing successfully
Markov Chains vs Hidden Markov Models
A common related concept is the Hidden Markov Model, or HMM.
The difference is relatively simple.
In a normal Markov Chain, the state is observable.
For example:
Server = Healthy
We can directly observe that state.
In a Hidden Markov Model, the underlying state is hidden and we observe signals generated by that state.
For example:
Hidden State
↓
Observed Measurements
A machine may have a hidden state such as:
Healthy / Degraded / Failing
while the system only observes:
Temperature, CPU utilization, vibration, latency, error rate
The hidden state must then be inferred from those observations.
HMMs have historically been used in areas such as speech recognition, biological sequence analysis, and natural language processing.
Markov Chains and Natural Language Processing
Markov models have also played an important role in the history of language modeling.
A simple Markov language model can estimate the probability of the next word based on previous words.
For example:
"I am learning..."
might lead to:
Python, Machine Learning, Data Engineering, ...
A simple model could learn probabilities for these transitions from a large collection of text.
However, traditional Markov models have a significant limitation: limited memory.
Language frequently depends on information much earlier in a sentence or document.
Modern Transformer models solve this problem much more effectively by using attention mechanisms to model relationships between tokens across a much larger context.
So while Markov Chains are not a replacement for modern Large Language Models, they are useful for understanding the evolution of probabilistic sequence modeling.
Markov Chains and PageRank
One famous application of Markov-style thinking is PageRank.
Imagine a user randomly navigating between web pages by following links.
The web can be represented as a graph:
Page A ───────► Page B
│ │
│ ▼
└────────────► Page C
Each page represents a state, and links represent transitions between states.
If we repeatedly model the movement of a hypothetical user across the web, we can calculate a long-term probability distribution representing how frequently different pages are visited under the model.
This idea became an important foundation of PageRank and demonstrates how Markov processes can be applied to large graphs.
How Transition Probabilities Are Calculated from Real Data
One of the most practical aspects of Markov Chains is that transition probabilities can be estimated directly from historical data.
Suppose our application records:
The total number of transitions from Search is:
8000+2000+5000=150008000 + 2000 + 5000 = 15000
Therefore:
P(Search→Product)=800015000=0.533P(Search \rightarrow Product) = \frac{8000}{15000} = 0.533
Similarly:
P(Search→Exit)=200015000=0.133P(Search \rightarrow Exit) = \frac{2000}{15000} = 0.133
and:
P(Search→Search)=500015000=0.333P(Search \rightarrow Search) = \frac{5000}{15000} = 0.333
These probabilities can then become one row of the transition matrix.
This is a very practical connection between data engineering and probabilistic modeling.
Markov Chains Are Not Always the Right Model
Although Markov Chains are useful, they make an important assumption.
The current state must contain enough information to reasonably predict the next state.
In real-world systems, this is not always true.
For example, suppose a customer's future purchase behavior depends on:
Their complete purchase history
Seasonality
Income
Promotions
Previous interactions
Competitor prices
Time since last purchase
Representing all of this information using a single state might lose important information.
In such cases, more sophisticated approaches may be appropriate.
These could include:
Machine learning models
Time-series models
Hidden Markov Models
Recurrent Neural Networks
LSTMs
Transformers
Survival analysis
Reinforcement learning
The right choice depends on the problem and the amount of information available.
Markov Chain vs Machine Learning
A Markov Chain and a machine-learning model solve related but different problems.
A Markov Chain can therefore be an excellent solution when the problem is naturally represented as states transitioning over time.
Common Mistakes When Learning Markov Chains
Mistake 1: Thinking Markov means completely random
A Markov Chain is probabilistic, but its behavior is governed by defined transition probabilities. Those probabilities can be learned from real historical data.
Mistake 2: Assuming the past never matters
The Markov property is an assumption about the model. If historical information contains important predictive information that is not represented by the current state, the model may lose accuracy.
Mistake 3: Confusing state probability with transition probability
These are different concepts.
A transition probability might be:
P(Rainy∣Sunny)=0.05P(Rainy|Sunny)=0.05
A state probability might be:
P(Rainy)=0.20P(Rainy)=0.20
The second depends on the overall state distribution and the transition structure.
Mistake 4: Assuming every Markov Chain reaches a stable state
Some Markov Chains do not converge to a unique stationary distribution. Properties such as irreducibility and aperiodicity are important when analyzing convergence.
Where Markov Chains Fit in Modern Data and AI
Markov Chains provide a useful conceptual bridge between probability and modern sequential modeling.
A simplified progression looks like:
Probability → Stochastic Processes → Markov Chains → Hidden Markov Models → Sequence Modeling → Deep Learning → Transformers
Modern AI systems are much more sophisticated than traditional Markov Chains, but the fundamental problem remains related: how do we model and reason about sequences of events?
For data engineers, this connection is especially valuable because data platforms increasingly process event streams.
A modern architecture might collect billions of events through Kafka, store historical data in an object store, transform it with Spark, load analytical data into a warehouse, and then use statistical or machine-learning models to extract behavioral patterns.
Markov Chains can be one of the simpler and more interpretable models in that broader architecture.
Key Takeaways
A Markov Chain models a system that moves between different states over time.
The central assumption is the Markov Property: the probability of the next state depends on the current state rather than the complete history.
The most important components are:
The most important equations to remember are:
πn=π0Pn\pi_{n} = \pi_0P^n
and for a stationary distribution:
π=πP\pi = \pi P
Conclusion
Markov Chains are a good example of how a relatively simple mathematical idea can be applied to complex real-world systems.
The key is to think about a problem in terms of states, transitions, and probabilities.
If you have a system where something continuously changes from one state to another—such as a customer moving through an application, a machine changing operational states, a user navigating a website, or an application generating sequential events—a Markov Chain can provide a clean way to understand that behavior.
For data engineers, the concept becomes even more practical when combined with modern data platforms. Event data collected through applications and streaming systems such as Kafka can be transformed into transition datasets, stored in data lakes or warehouses, and then used to calculate transition probabilities and long-term behavioral patterns.
Markov Chains may not be the most sophisticated model available today, but they remain an important foundation for understanding probabilistic sequence modeling.
And sometimes, a simple and interpretable model is exactly what a problem needs.