Introduction 📚

This article explores the fundamental concepts of information theory, which form the mathematical foundation for many machine learning algorithms. Understanding these concepts is crucial for grasping how models process and learn from data.

Information Quantity

When an event A occurs with probability P(A), the information quantity I(A) measures how much information we gain from observing this event:

$ I(A) = -\log P(A)$

Key insight: Rare events carry more information than common ones. This makes intuitive sense - learning that a rare event occurred tells us more than learning about a common event.

Properties of Information Quantity

  • Inversely proportional to probability: Lower probability events provide more information
  • Additive for independent events: When events A and B are independent, their combined information is the sum of individual information quantities

$ I(A \cap B) = I(A) + I(B)$

Entropy

Entropy H(A) represents the expected amount of information from a probability distribution:

$ H(A) = -P(A)\log P(A)$

Think of entropy as a measure of uncertainty or “surprise” in a system. Higher entropy means more uncertainty and more information potential.

Cross Entropy

Cross entropy measures the difference between two probability distributions p(x) and q(x):

$ H(p,q) = -\sum_{x} p(x)\log q(x)$

This is particularly important in machine learning, where we often want to compare our model’s predicted distribution (q) with the true distribution (p).

Kullback-Leibler Divergence

KL divergence quantifies how much one probability distribution differs from another:

$ D_{KL}(p||q) = \sum_{x} p(x)\log \frac{p(x)}{q(x)}$

Key Properties:

  • Asymmetric: D_KL(p||q) ≠ D_KL(q||p)
  • Non-negative: Always ≥ 0
  • Zero when identical: D_KL(p||q) = 0 when p(x) = q(x)

Mathematical Relationships 🔗

These three concepts are interconnected through a fundamental relationship:

$ H(p,q) = H(p) + D_{KL}(p||q)$

This equation shows that cross entropy equals the entropy of the true distribution plus the KL divergence between the true and predicted distributions.

Practical Applications

  • Loss functions in classification tasks
  • Model evaluation and comparison
  • Feature selection and dimensionality reduction
  • Regularization techniques in deep learning