ML Notes Mathematics FundamentalsProbability

Likelihood, Log-Likelihood, and MLE

Math fundamentals note for ML preparation.

16 February 2026

Contents

Core Idea

Likelihood treats parameters as unknowns and observed data as fixed.
MLE picks parameters that maximize observed-data likelihood.

Key Formula

\hat\theta=\arg\max_\theta \prod_{i=1}^n p(x_i\mid\theta)

Usually optimize:

\hat\theta=\arg\max_\theta \sum_{i=1}^n \log p(x_i\mid\theta)

Why This Matters for ML

MLE is the statistical foundation of many classical ML estimators.
Log-likelihood turns products into sums, enabling stable optimization and gradient methods.
Logistic regression and many generative models are trained by maximizing likelihood objectives.
Understanding likelihood is key for comparing models and defining principled losses.

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