Statistics How To

Kullback–Leibler KL Divergence

Statistics Definitions >

Kullback–Leibler divergence (also called KL divergence, relative entropy information gain or information divergence) is a way to compare differences between two probability distributions p(x) and q(x). More specifically, the KL divergence of q(x) from p(x) measures how much information is lost when q(x) is used to approximate p(x). It answers the question: If I used the “not-quite” right distribution q(x) to approximate p(x), how many bits of information do I need to more accurately represent p(x)?

KL Divergence Formula

The formula for the divergence of two discrete probability distributions, defined over a random variable x ∈ X, is as follows:


  • X is the set of all possible variables for x.

The log function is sharp close to zero, so this may allow sensitive detection of small probability distribution changes (Sugiyama, 2015).

For continuous probability distributions, the formula (Tyagi, 2018) involves integral calculus:

KL Distance

KL divergence is sometimes called the KL distance (or a “probabilistic distance model”), as it represents a “distance” between two distributions. However, it isn’t a traditional metric (i.e. it isn’t a unit of length). Firstly, it isn’t symmetric in p and q; In other words, the distance from P to Q is different from the distance from Q to P. In addition, it doesn’t satisfy the triangle inequality (Manning & Schütze, 1999).


Ganascia, J. et al. (2008). Discovery Science: 15th International Conference, DS 2012, Lyon, France, October 29-31, 2012, Proceedings. Springer.
Han, J. (2008). Kullback-Leibler Divergence. Retrieved March 16, 2018 from:
Kullback, S. & Liebler, R. (1951). On Information and Sufficiency. Annals of Mathematical Statistics. 22(1): 79-86.
Manning, C. and Schütze, H. (1999). Foundations of Statistical Natural Language Processing. MIT Press.
Sugiyama, M. (2015). Introduction to Statistical Machine Learning. Morgan Kaufmann.
Tyagi, V. (2018). Content-Based Image Retrieval: Ideas, Influences, and Current Trends. Springer.


Need help with a homework or test question? With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!

Statistical concepts explained visually - Includes many concepts such as sample size, hypothesis tests, or logistic regression, explained by Stephanie Glen, founder of StatisticsHowTo.

Comments? Need to post a correction? Please post a comment on our Facebook page.

Check out our updated Privacy policy and Cookie Policy

Leave a Reply