How Gaussian Distribution Maximizes Entropy — The Proof

The normal distribution, often referred to as the Gaussian distribution, stands as a fundamental element in the realm of statistics and probability theory due to its inherent mathematical properties and widespread applicability. One of its most compelling attributes is the ability to maximize entropy given specific constraints on mean and variance. In this blog I aim to explore the mathematical proof, employing the calculus of variations to delve into the depths of why the normal distribution is so prevalent and powerful.

(You can read the previous blog which provides an elaborate mathematical perspective on the properties of the Gaussian distribution here: The Gaussian Distribution: A Mathematical Perspective)

Understanding Entropy in Probability Distributions

Entropy, a concept originally rooted in thermodynamics, has significant implications in information theory and statistics. In the context of probability distributions, entropy measures the uncertainty or unpredictability associated with a random variable. For a continuous random variable 𝑋 with a probability density function 𝑝(𝑥), the entropy 𝐻 is defined mathematically as:

This expression provides a quantitative measure of the amount of “information” or “randomness” inherent in the distribution of 𝑋.

The Problem of Entropy Maximization

The entropy maximization problem asks: Among all probability distributions with a given mean 𝜇 and variance 𝜎2, which distribution maximizes the entropy 𝐻? This is a classic problem in information theory and statistical mechanics, reflecting the principle of maximum entropy as a method of inference.

Setting Up the Maximization

To address this problem, we approach it through a functional optimization framework. We are tasked with maximizing the entropy functional:

subject to constraints on the normalization condition:

The mean constraint:

and the variance constraint:

These constraints ensure that the probability distribution 𝑝(𝑥) is properly normalized, has the correct mean 𝜇, and the specified variance 𝜎2. These are fundamental conditions that must be met in the analysis of any probability distribution, particularly when discussing properties like entropy.

Utilizing Calculus of Variations

The calculus of variations provides the tools needed to solve this constrained optimization problem. We formulate a Lagrangian incorporating the constraints with Lagrange multipliers 𝜆0​, 𝜆1​, and 𝜆2:

Derivation and Solving for 𝑝(𝑥)

The first variation of 𝐿 with respect to 𝑝 leads to the condition for an extremum. Setting the derivative equal to zero, we find:

This leads us to express 𝑝(𝑥) as:

Applying the constraints simplifies the multipliers, giving:

Therefore, the probability density function becomes:

To finalize 𝜆0​, use the normalization condition:

Solving this, you find that 𝜆0 corresponds to the normalization factor for the Gaussian distribution, confirming that:

This derivation conclusively shows that the normal distribution maximizes entropy under the constraints of given mean and variance, exemplifying its unique property of distributing probability mass in the most “uncertain” way possible under these conditions.

Broader Implications and Insights

The proof not only establishes why the normal distribution is a natural choice in many statistical applications but also illustrates the connection between entropy and statistical inference. This result is pivotal for understanding the behavior of complex systems and underlies many principles in areas ranging from physics to finance.