cross-entropy, KL divergence, and Shannon entropy?

 begin quotes:

Shannon Entropy measures the inherent uncertainty in a single probability distribution. Cross-Entropy measures the difference between two distributions, often used as a loss function in machine learning. KL Divergence calculates the "extra" information wasted when approximating a true distribution with a predicted one. 

These three fundamental concepts from information theory are deeply interconnected mathematically, with each serving a specific purpose in evaluating and comparing probability distributions. 

1. Shannon Entropy

Shannon Entropy quantifies the uncertainty, surprise, or information content inherent in a single probability distribution

. 

• Concept: If an event is highly likely to happen, observing it yields very little information (low surprise, low entropy). If outcomes are highly unpredictable, the entropy is high.
• Formula:
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   

2. Cross-Entropy

Cross-Entropy measures the average number of bits needed to identify an event drawn from the true distribution

, but using a wrong, predicted distribution

instead. 

• Concept: It is commonly used as a cost/loss function in machine learning classification models (e.g., logistic regression, neural networks). The model aims to adjust its predicted distribution
  
  to match the true distribution
  
  , which minimizes the cross-entropy.
• Formula:
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
   

3. Kullback-Leibler (KL) Divergence

Also known as relative entropy, KL Divergence measures how much one probability distribution

diverges from an expected or true distribution

. 

• Concept: It tells you how much information is lost if you use
  
  to approximate
  
  . It is not a true distance metric because it is asymmetric (i.e.,
  
  
  
  
  
  
  
  
  
  
  ).
• Formula:
  
   

The Mathematical Connection

These three concepts are fundamentally tied together by a single equation: 

What this means in practice:
When evaluating a machine learning model, minimizing the cross-entropy loss

is perfectly equivalent to minimizing the KL divergence

. This is because the entropy of the true distribution

remains a constant, immovable value. 

To explore the mathematical derivations and real-world implementations, consult the detailed information theory breakdowns on Wikipedia's Kullback-Leibler Divergence page and Cross-Validated's Statistics Stack Exchange.

• What is the difference between Cross-entropy and KL ...

  Jul 19, 2018 — where the first term of the right hand side is the entropy of distribution A, the second term can be interpreted as the expectatio...

  

  Stack Exchange
• Shannon Entropy and Kullback-Leibler Divergence

  Page 1. Chapter 28. Shannon Entropy and. Kullback-Leibler. Divergence. Section 28.1 introduces Shannon entropy and its most basic ...

  

  Carnegie Mellon University
• [D] A Short Introduction to Entropy, Cross-Entropy and KL- ...

  Feb 5, 2018 — of course you know that a bit is a number that is either equal to zero or one. but not all bits are useful some of them are redund...

  

  9m

  

  Reddit·Aurélien Géron

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Shannon Entropy and Kullback-Leibler Divergence

Carnegie Mellon University

https://www.stat.cmu.edu › notes › lecture-28

PDF

Section 28.1 introduces Shannon entropy and its most basic prop- erties, including the way it measures how close a random variable is.

8 pages

Information Theory Fundamentals: Entropy, Cross ...

Nima Sarang

https://nimasarang.com › blog › 2024-08-24-informati...

Aug 24, 2024 — A guide to information theory covering entropy, cross-entropy, KL divergence, and Jensen-Shannon divergence with proofs, examples, and ...

What is the difference between Cross-entropy and KL ...

Stack Exchange

https://stats.stackexchange.com › questions › what-is-th...

Jul 19, 2018 — Both the cross-entropy and the KL divergence are tools to measure the distance between two probability distributions, but what is the difference between ...

4 answers  ·  Top answer: You will need some conditions to claim the equivalence between minimizing cross entropy ...

Relationship between KL divergence and entropy

2 answers

May 9, 2020

Why KL-Divergence uses "ln" in its formula? - Cross ...

2 answers

Oct 13, 2014

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Understanding Shannon Entropy and KL-Divergence ...

Medium · Priyanshu maurya

80+ likes · 1 year ago

In this article, we will focus on entropy from the information theory perspective to better understand its role in the context of machine learning.

Cross-entropy and Shannon entropy

machinelearningreference.com

https://machinelearningreference.com › 10-foundations

Dec 23, 2024 — Since the KL divergence is the difference between the cross-entropy and the Shannon entropy, we conclude that cross-entropy can never be less ...