gen-in-3-4-sentences-explain-maximum-likelihood-estimation-state-ede2

In 3-4 sentences, explain maximum likelihood estimation. State any key formula.

Maximum likelihood estimation (MLE) is a method for estimating the parameters of a statistical model by finding the parameter values that maximize the likelihood of observing the given data. The likelihood function L(θ|data) measures how probable the observed data is for different parameter values θ, and MLE seeks the θ that makes this likelihood as large as possible. The key formula is: **θ̂_MLE = argmax_θ L(θ|x₁, x₂, ..., xₙ)** or equivalently **θ̂_MLE = argmax_θ log L(θ|x₁, x₂, ..., xₙ)**, where the log-likelihood is often used for computational convenience. For independent observations, the likelihood is the product of individual probability densities: L(θ) = ∏ᵢ f(xᵢ|θ).