How to Calculate Kullback-Leibler Divergence in Python

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In information theory and statistics, the Kullback-Leibler (KL) divergence, also known as relative entropy, is a measure of how one probability distribution diverges from a second, expected probability distribution. It is used extensively in machine learning and data science, for tasks such as natural language processing, pattern recognition, and anomaly detection.

In this article, we will explain how to calculate the KL divergence using Python, one of the most popular programming languages in the data science community.

What is Kullback-Leibler Divergence?

Kullback-Leibler divergence is a non-symmetric measure of the difference between two probability distributions P and Q. In simpler terms, it quantifies how much one distribution differs from another.

Mathematically, the KL divergence of Q from P is defined for discrete distributions as:

D_{KL}(P || Q) = Σ P(i) * log(P(i) / Q(i))

And for continuous distributions as:

D_{KL}(P || Q) = ∫ P(x) * log(P(x) / Q(x)) dx

Remember that KL divergence is not symmetric, meaning that D_{KL}(P || Q) is not equal to D_{KL}(Q || P). It’s also non-negative and equals zero if and only if P and Q are the same distribution in the case of absolute continuity.

How to Calculate KL Divergence in Python

Python provides several libraries that make it easy to calculate KL divergence. One of the most common methods is by using SciPy and NumPy.

First, let’s import these libraries:

import numpy as np
from scipy.special import kl_div

Now let’s create two simple discrete probability distributions:

# Probability distribution P
P = np.array([0.1, 0.2, 0.7])

# Probability distribution Q
Q = np.array([0.2, 0.2, 0.6])

We can calculate the KL divergence from P to Q as follows:

# Calculate KL Divergence
kl_PQ = kl_div(P, Q).sum()

In this case, kl_div(P, Q) computes the KL divergence for each pair of corresponding elements in P and Q, and sum() adds up these values to compute the total divergence.

Dealing with Zero Probabilities

One issue with the KL divergence is that it is not defined when Q(i) is zero for any i where P(i) is not zero, since the logarithm of zero is undefined.

To handle this in Python, one common approach is to add a small constant to each probability. This is known as Laplace smoothing or add-one smoothing:

# Add a small constant to P and Q
P += 0.00001
Q += 0.00001

This will prevent any division by zero errors when calculating the KL divergence.

Calculating KL Divergence for Continuous Distributions

In the case of continuous distributions, we can’t calculate the KL divergence directly from the probability density functions. We need to use numerical integration.

For instance, if we want to calculate the KL divergence between two Gaussian distributions, we first need to define the Gaussians. We can use scipy.stats.norm for this:

from scipy.stats import norm

# Define two Gaussian distributions
P = norm(loc=0, scale=1)
Q = norm(loc=1, scale=2)

We can then calculate the KL divergence by defining the integral as a function and using scipy.integrate.quad to compute the integral:

from scipy.integrate import quad

def kl_continuous(P, Q, lower=-np.inf, upper=np.inf):
    # Define the integrand
    def integrand(x):
        return P.pdf(x) * np.log(P.pdf(x) / Q.pdf(x))

    # Calculate the integral
    return quad(integrand, lower, upper)[0]

# Calculate KL Divergence
kl_PQ = kl_continuous(P, Q)

This code first defines an integrand function, which represents the integrand of the KL divergence formula. It then computes the integral of this function over the range of the distribution using quad. The [0] at the end of quad is used to get the integral value, since quad returns both the value of the integral and an estimate of the error.

Applications of KL Divergence

KL Divergence has many practical applications in data science and machine learning. Here are a few examples:

  • In natural language processing, KL divergence can be used to determine the similarity between two text documents, by treating them as probability distributions of words.
  • In machine learning, KL divergence is used in algorithms such as t-SNE and variational autoencoders.
  • KL divergence can also be used for anomaly detection, by comparing a suspected anomaly to a normal distribution of data.


KL Divergence is a powerful tool for comparing probability distributions, and Python makes it easy to calculate and apply in practical applications. By using the built-in functions in libraries like SciPy and NumPy, you can calculate KL divergence for both discrete and continuous distributions in just a few lines of code.

Understanding KL divergence and other measures of distribution similarity can greatly aid you in your data science journey. They offer key insights into your data and allow you to build more effective and accurate models.

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