Maximum Likelihood Estimation Example: Fitting a Normal Distribution with Data

2025-11-28 · Source: Steve Brunton · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Data Science & Analytics · Depth: Intermediate, long

Summary

This content details the application of Maximum Likelihood Estimation (MLE) to estimate the parameters of a normal distribution from a given dataset. It begins by defining the likelihood function and explaining the rationale for using its logarithm for optimization. The core derivation involves constructing the joint probability density function for independent and identically distributed (IID) normal variables, then transforming it into a log-likelihood function. The process explicitly calculates the partial derivatives of the log-likelihood function with respect to the mean (mu) and variance (sigma squared). By setting these partial derivatives to zero, the MLE estimates for mu and sigma squared are derived, revealing that the optimal mu is the sample mean (x-bar) and the optimal sigma squared is the sample variance (1/n * sum((xi - x-bar)^2)). These results are consistent with estimates obtained via the method of moments.

Key takeaway

For Data Scientists and Machine Learning Engineers working with normally distributed data, understanding MLE is crucial. Your optimal estimates for the mean and variance will align with the sample mean and sample variance, respectively. This method provides a robust statistical foundation for parameter estimation, which is broadly applicable, even in complex models like neural networks. Practice deriving these estimates for other distributions to solidify your understanding.

Key insights

MLE provides optimal parameter estimates for a probability distribution given observed data by maximizing the log-likelihood function.

Principles

• IID data simplifies joint PDF to a product of individual PDFs.
• Logarithms convert products into sums, simplifying optimization.
• Setting partial derivatives to zero finds optimal parameters.

Method

Define the joint PDF, take its logarithm to form the log-likelihood function, then compute and set partial derivatives with respect to parameters to zero to solve for optimal estimates.

In practice

• Use MLE to estimate parameters for various distributions.
• Apply to neural networks where weights are distribution parameters.
• Verify maxima by computing second derivatives.

Topics

• Maximum Likelihood Estimation
• Normal Distribution
• Parameter Estimation
• Log-Likelihood Function
• Method of Moments

Best for: AI Student, Data Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by Steve Brunton.