Difference Between Entropy and Information Gain

Entropy and information gain are key concepts in domains such as information theory, data science, and machine learning. Information gain is the amount of knowledge acquired during a certain decision or action, whereas entropy is a measure of uncertainty or unpredictability. People can handle difficult situations and make wise judgments across a variety of disciplines when they have a solid understanding of these principles. Entropy can be used in data science, for instance, to assess the variety or unpredictable nature of a dataset, whereas Information Gain can assist in identifying the qualities that would be most useful to include in a model. In this article, we'll examine the main distinctions between entropy and information gain and how they affect machine learning.

What is Entropy?

The term "entropy" comes from the study of thermodynamics, and it describes how chaotic or unpredictable a system is. Entropy is a measurement of a data set's impurity in the context of machine learning. In essence, it is a method of calculating the degree of uncertainty in a given dataset.

The following formula is used to compute entropy −

$$\mathrm{Entropy(S) = -p1log2p1 - p2log2p2 - … - pnlog2pn}$$

S is the data set, and p1 through pn are the percentages of various classes inside the data. The resultant entropy value is expressed in bits since the base 2 logarithm used in this method is typical.

Consider a dataset with two classes, A and B, in order to comprehend this formula. The entropy can be determined as follows if 80% of the data is in class A and 20% is in class B −

$$\mathrm{Entropy(S) = -0.8log20.8 - 0.2log20.2 = 0.72 bits}$$

This indicates that the dataset is impurity-rich, with an entropy of 0.72 bits.

What is Information Gain?

Information Gain is a statistical metric used to assess a feature's applicability in a dataset. It is an important idea in machine learning and is frequently utilized in decision tree algorithms. By contrasting the dataset's entropy before and after a feature is separated, information gain is estimated. A feature's relevance to the categorization of the data increases with information gain.

When the dataset has been divided based on a feature, information gain calculates the entropy decrease. The amount of knowledge a feature imparts about the class is measured by this metric. Selecting the characteristic that provides the most information about the class will help you achieve your aim of maximizing information gain.

The following formula is used to compute information gain −

$$\mathrm{Information Gain(S, A) = Entropy(S) – ∑ (|Sv| / |S|) * Entropy(Sv)}$$

The number of elements in Sv is given by |Sv|, where S is the set of data, A is a feature, Sv is the subset of S for which feature A takes the value v, and S is the total number of elements in S.

Think of a dataset with two characteristics, X and Y, to better comprehend this formula. The information gain can be calculated as follows if the data is to be divided based on characteristic X −

$$\mathrm{Information Gain(S, X) = Entropy(S) – [(3/5) * Entropy(S1) + (2/5) * Entropy(S2)]}$$

where S1 is the subset of data where feature X takes a value of 0, and S2 is the subset of data where feature X takes a value of 1. These two subsets' entropies, Entropy(S1) and Entropy(S2), can be determined using the formula we previously covered.

The amount by which the dataset will be divided based on characteristic X will be shown by the information gain that results.

Key Differences between Entropy and Information Gain


Information Gain

Entropy is a measurement of the disorder or impurity of a set of occurrences. It determines the usual amount of information needed to classify a sample taken from the collection.

Information gain is a metric for the entropy reduction brought about by segmenting a set of instances according to a feature. It gauges the amount of knowledge a characteristic imparts to the class of an example.

Entropy is calculated for a set of examples by calculating the probability of each class in the set and using that information in the entropy calculation.

By dividing the collection of instances depending on the feature and calculating the entropies of the resulting subsets, information gain is determined for each feature. The difference between the entropy of the original set and the weighted sum of the entropies of the subsets is thus the information gain.

Entropy quantifies the disorder or impurity present in a collection of instances and aims to be minimized by identifying the ideal division.

By choosing the feature with the maximum information gain, the objective of information gain is to maximize the utility of a feature for categorization.

Entropy is typically taken into account by decision trees for determining the best split.

Decision trees frequently employ information gain as a criterion for choosing the optimal feature to split on.

Entropy usually favors splits that result in balanced subgroups.

Splits that produce imbalanced subsets with pure classes are frequently preferred by information gain.

Entropy can control continuous characteristics by discretizing them into bins.

By choosing the split point that maximizes the information acquisition, continuous features may also be handled.

Calculating probabilities and logarithms, which can be computationally costly, is necessary to determine entropy.

Entropies and weighted averages must be calculated in order to gather information, which can be computationally costly.

Entropy is a versatile indicator of impurity that may be applied to a variety of classification issues.

For binary classification issues, information gain is a particular measure of feature usefulness that works well.

Entropy, which is given in bits, calculates the typical amount of data required to categorize an example.

Information gain, which is also stated in bits, indicates the reduction in uncertainty attained by splitting based on a feature.

If there are too many characteristics or the tree is too deep, entropy might result in overfitting.

If the tree is too deep or there are too many irrelevant characteristics, information gain may potentially result in overfitting.


Entropy and information gain are two fundamental, interrelated ideas in machine learning. Although information gain evaluates the uncertainty reduction that can be accomplished by segmenting the data on a certain attribute, entropy measures the impurity or uncertainty of a dataset. While creating a decision tree, the optimum feature for segmenting a dataset is chosen using information gain, which is calculated using entropy. Building precise and efficient machine learning models requires an understanding of the distinctions between these two ideas.

Updated on: 25-Apr-2023

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