Why Mutual Information Deserves More Love Than It Gets

Author(s): Ashwin Biju Alikkal

Originally published on Towards AI.

✨ Introduction

A few months into working on ML projects, I was pretty confident in my ability to identify useful features. Correlation matrices? Check. Feature heatmaps? Double check. But then something strange happened — a feature with zero correlation significantly impacted my model’s accuracy. That’s when I stumbled upon Mutual Information, and it changed how I understood relationships in data forever.

Most tutorials I had followed until then praised correlation, but they barely whispered about Mutual Information (MI). This post is my attempt to fix that. We’ll look at how MI works, why it’s often more insightful than correlation, and we’ll dive into a bit of the math that makes it so powerful.

If you’ve ever wondered why some features matter even when they seem unrelated at first glance — this one’s for you.

🗂️ Index

1. Correlation: The Usual Suspect
2. Mutual Information: The Hidden Gem
3. Mutual Information vs Correlation — A Mathematical Example
4. When Should You Use MI Instead of Correlation?
5. Key Takeaways

I’ll be sharing the necessary Python codes and outputs for your understanding in between.

1. Correlation: The Usual Suspect

Correlation measures the linear relationship between two variables. It answers the question: If X increases, what happens to Y?

Mathematically, the Pearson Correlation Coefficient ρ is defined as:

So if the relationship is non-linear, correlation might totally miss it.

Hence enters….

2. Mutual Information: The Hidden Gem

Mutual Information measures the shared information between two variables. It tells you how much knowing one variable reduces the uncertainty of the other.

So if X and Y are random variables, the Mutual Information is defined as:

This measures how different the joint distribution P(X = x, Y = y) is from the product of the marginal distributions P(x)*P(y).

• If X and Y are independent, I(X; Y) = 0.
• If one predicts the other perfectly, MI is maximised.

3. Mutual Information vs Correlation — A Mathematical Example

Now, let us bring this debate to life with two contrasting examples. We’ll look at how correlation and mutual information behave when faced with a linear and a non-linear relationship.

We’ll generate both examples synthetically so you can run them easily in your own notebook.

import numpy as np
import pandas as pd
from sklearn.feature_selection import mutual_info_regression
import matplotlib.pyplot as plt

# Linear data with noise
X1 = np.linspace(0, 100, 1000)
y1 = 2 * X1 + np.random.normal(0, 10, 1000)

df1 = pd.DataFrame({'X1': X1, 'y1': y1})

# Calculate correlation
corr1 = df1.corr().loc['X1', 'y1']

# Mutual Information
mi1 = mutual_info_regression(df1[['X1']], df1['y1'])[0]

print(f"[Linear] Correlation: {corr1:.4f}")
print(f"[Linear] Mutual Information: {mi1:.4f}")

plt.scatter(df1['X1'], df1['y1'], alpha=0.5)
plt.title("Linear: High Correlation & High MI")
plt.xlabel("X1")
plt.ylabel("y1")
plt.show()

In the above image, the correlation is 0.98 and MI is 1.71. So we can say that X1 is linearly dependent on Y1. It’s a kind of pattern that Pearson Correlation loves and MI agrees with.

Hence, we say that both metrics capture the dependency correctly, i.e. the stronger the linear trend, the higher both values will be.

Now let’s take the example of a sine wave and add some noise.

import numpy as np
import pandas as pd
from sklearn.feature_selection import mutual_info_regression
import matplotlib.pyplot as plt

# Generate sine data
X = np.linspace(0, 10, 1000).reshape(-1, 1)
y = np.sin(X).ravel() + 0.1 * np.random.randn(1000) # Non-linear + some noise

# DataFrame for correlation
df = pd.DataFrame({'X': X.ravel(), 'y': y})

# Calculate correlation
correlation = df.corr().loc['X', 'y']

# Calculate mutual information (continuous)
mi = mutual_info_regression(X, y)[0]

print(f"Correlation: {correlation:.4f}")
print(f"Mutual Information: {mi:.4f}")

plt.scatter(X, y, alpha=0.4, edgecolors='k')
plt.title("Sine Relationship (Continuous X and y)")
plt.xlabel("X")
plt.ylabel("y = sin(X) + noise")
plt.grid(True)
plt.show()

Now, in the above plot, we can see the sine wave along with some noise. Here, the correlation was -0.0644 (approximately 0). But the real deal is MI, which came out to be 1.7302.

So here correlation thinks the variables are unrelated but MI finds it otherwise.

What does this teach us?

• Correlation is narrow-minded — it only sees straight lines.
• Mutual Information is open-minded — it sees structure, even if it’s curvy, wavy, or weird.

So if your data isn’t playing by textbook rules, MI is your best friend. It listens when correlation doesn’t.

4. When Should You Use MI Instead of Correlation?

5. Key Takeaways

Here’s what I wish I knew earlier — and what you should take with you:

• Use both together — Correlation gives you fast intuition. Mutual Information gives you deep insight. When used side by side, they reveal much more than either alone.
• Use Mutual Information when your data isn’t simple — If you’re dealing with tree-based models, non-linear patterns, or a mix of categorical and continuous variables, MI is your best bet.
• Stick to Correlation when you’re exploring linear trends — It’s fast, directional, and perfect for early-stage EDA.
• Don’t be scared of MI — It might feel unfamiliar at first, but all it’s doing is asking how much knowing one thing helps you guess another.
• Always visualise your data — Even if numbers confuse you, your eyes won’t. MI caught the sine wave. So did we.

Mutual Information is like that quiet genius in the room — not flashy, not obvious, but incredibly powerful when you understand how to work with it. So the next time your data feels confusing or your model ignores a seemingly important feature, ask yourself — have you gone beyond correlation?

Because correlation is easy to see. But mutual information helps you truly understand.

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Published via Towards AI