Mutual Information: How ideas from information theory influence machine learning

Author(s): Mahmoud Abdelaziz, PhD

Originally published on Towards AI.

What Is Mutual Information?

Mutual information is a concept from information theory that tells us how much one variable X tells us about another Y. In simpler terms, it’s a way to measure how much knowing the value of one thing reduces our uncertainty about something else.

In other words, mutual information compares how often X and Y occur together with how often they would occur if they were completely unrelated. If knowing X gives us useful information about Y, the mutual information will be greater than zero.

For example, if you know whether it’s raining (X), that might help you guess whether someone is carrying an umbrella (Y). If the two are closely connected, mutual information will be high. If they’re unrelated, mutual information will be zero. In probabilistic terms, this will occur when the joint probability between X and Y equals the product of the marginal probabilities in the above formula, thus resulting in zero mutual information.

Unlike other tools that only work in special cases, mutual information is incredibly flexible. It can detect all kinds of relationships — linear, curved, periodic, or even irregular ones that are hard to describe in simple terms.

Why Is This Better Than Correlation?

Most people have heard of correlation, especially Pearson’s correlation, which measures how well two variables follow a straight line. But here’s the problem: not all relationships are straight lines.

Imagine a machine that only fails when the temperature is too high or too low, but not in the middle. This creates a U-shaped pattern. A simple correlation might say there’s no connection between temperature and failure rate because the high and low points cancel each other out. But mutual information still picks it up, because it sees that both extremes are important — even if the middle isn’t.

This makes mutual information more powerful when we’re dealing with complex systems, where the relationships between things aren’t always obvious.

How Bayesian Networks Use Mutual Information

Bayesian networks are graphs that show how different variables are related. Each node in the graph represents a variable, like temperature, pressure, or humidity. The arrows between nodes represent influence — one variable affecting another.

When machines learn the structure of a Bayesian network from data, they need to figure out which variables are connected and which are not. This is where mutual information becomes essential.

At first, we might assume that all variables are connected. We then calculate the mutual information between every pair of variables. If the mutual information between two variables is high, it suggests there’s a direct link between them — they share important information.

But appearances can be deceiving. Sometimes, two variables seem to be related, but it’s only because they’re both influenced by a third variable. For instance, ice cream sales and sunburn rates may look connected, but both are actually caused by hot weather.

To uncover these hidden influences, we use a more refined tool called conditional mutual information. It measures the relationship between two variables after accounting for a third. If the conditional mutual information drops to zero, it means that third variable explains away the connection.

This step helps the machine “filter out the noise” and focus only on the true direct links. It’s like tuning a radio — removing background interference so you can clearly hear the actual broadcast.

Example: A Factory Case Study

Imagine a factory that produces metal parts. The quality of each part depends on the machine settings and also on the humidity in the air.

At first, we might see that machine settings and product quality are closely related. Mutual information between them is high. But when we also consider humidity, we find that this connection weakens or even disappears. Why? Because changes in humidity are actually affecting both the settings and the quality.

By using conditional mutual information, we can discover that humidity was the real cause behind the changes. This helps us avoid wrong conclusions and build a more accurate model of the system.

Bayesian Networks as Communication Systems

There’s another powerful way to think about Bayesian networks: like communication systems. This idea comes from Claude Shannon, the father of information theory.

In a communication system, you have a sender, a message, a channel, and a receiver. The sender tries to send a message, but the channel might add noise or distortion. Mutual information tells us how much of the original message makes it through.

In the same way, each variable in a Bayesian network “sends” information to the others through the connections in the graph. The amount of information that gets through — how much one variable tells us about another — is measured by mutual information. And just like a noisy communication channel, uncertainty in the data can affect how much we can learn.

When we learn a Bayesian network from data, it’s like trying to reverse-engineer this communication system. We ask: Who is sending messages to whom? How strong are those messages? Where is the noise coming from? Mutual information helps us answer all of these questions.

When Bayesian and Frequentist Statistical Inference Frameworks meet

Despite mutual information being one of the most powerful tools in both statistics and information theory, many forget that computing it from data is a statistical act. When we estimate mutual information from finite samples, we’re dealing with a random quantity. We need to ask: Is this mutual information statistically significant?

This is where hypothesis testing — from the frequentist toolbox — comes in. We set up a null hypothesis that the true mutual information (MI) between two variables is zero, and then compute an empirical MI from the data. If this observed MI is sufficiently large under the sampling distribution of the null, we reject the hypothesis — concluding that the variables are likely dependent.

Interestingly, this entire process lives at the intersection of Bayesian and frequentist thinking. While mutual information itself is deeply rooted in the Bayesian view of uncertainty and belief propagation (especially in Bayesian networks), its practical evaluation often requires a frequentist frame. The test for significance helps us avoid overfitting in structure learning, prune spurious edges, and ensure that what we interpret as “informative” is not just a product of noise.

This blend of paradigms isn’t a contradiction — it’s a synergy. In modern machine learning and data science, the best insights often come from respecting both perspectives. Mutual information is just one example of how Bayesian structure and frequentist inference can work together to reveal what really matters in data.

A Real-World Example: Resource Allocation in Wireless Sensor Networks

Imagine a wireless sensor network deployed in a smart agriculture system, where each sensor node measures variables like soil moisture, temperature, and humidity. These nodes communicate wirelessly with a central controller to optimize irrigation and fertilization in real-time. In such a network, mutual information plays a crucial role in deciding which sensor data should be prioritized over limited-bandwidth wireless links.

If two sensors provide highly redundant data — say, both are measuring the same environmental trend — their mutual information will be high. This redundancy allows the system to reduce transmission from one node without significant loss of information at the controller. On the other hand, if a sensor provides unique insights (i.e., it has high mutual information with the target variable but low mutual information with other nodes), it becomes critical to transmit its data reliably.

Here, mutual information helps solve both a structure learning problem (which sensors are conditionally independent given others) and a communication problem (how to allocate bandwidth and power to maximize useful information flow). This dual role shows how concepts from Bayesian networks and communication theory naturally converge in real-world wireless applications.

Learning as Decoding Hidden Messages

The more we think about it, the more learning starts to look like communication.

Each variable in our dataset is like someone trying to send a message. But the message is incomplete, noisy, or hard to understand. Learning a model — like a Bayesian network — is about figuring out who’s sending messages to whom, and what those messages mean.

When data is missing, or when the connections are fuzzy, it’s like trying to listen to a weak radio signal. We use tools like mutual information to tune in, reduce the noise, and discover the underlying structure of the system.

This view — seeing learning as a form of decoding — opens up exciting new ways to combine ideas from machine learning and information theory. Concepts like error correction, compression, and communication capacity could help us build smarter and more reliable models.

Conclusion: Why Mutual Information Matters

Mutual information is much more than just a mathematical tool. It’s a new way of thinking about learning, modeling, and understanding the world.

It helps us see patterns that other tools miss. It helps us build better models, especially when the relationships between variables are complex or hidden. And it gives us a shared language that connects statistics, computer science, and communication theory.

As data science continues to grow, mutual information will become a central idea — one that reminds us that learning isn’t just about numbers. It’s about discovering meaning, reducing uncertainty, and decoding the messages hidden inside the data.

Suggested Reading

• Cover, T. & Thomas, J. (2006). Elements of Information Theory

• Spirtes, P., Glymour, C., & Scheines, R. (2000). Causation, Prediction, and Search

• Kraskov, A., Stögbauer, H., & Grassberger, P. (2004). Estimating Mutual Information

• MacKay, D. (2003). Information Theory, Inference, and Learning Algorithms

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Published via Towards AI