Normallikelihood

Normallikelihood, commonly referred to as the Gaussian likelihood, is the likelihood function used when data are assumed to be drawn from a normal distribution with unknown parameters. It expresses how probable the observed data are as a function of those parameters, under the assumption of independent and identically distributed observations.

For a sample x1, x2, ..., xn from a normal distribution with mean μ and standard deviation σ (or variance σ^2), the normal likelihood is L(μ, σ^2; x) = ∏_{i=1}^n (1/(√(2π) σ)) exp(-(xi - μ)^2 / (2σ^2)). This can be written more compactly as L = (2π σ^2)^(-n/2) exp(-∑(xi - μ)^2 / (2σ^2)). The log-likelihood is often used in practice: l(μ, σ^2) = -n/2 log(2π) - n log σ - (1/(2σ^2)) ∑(xi - μ)^2.

Maximum likelihood estimation with normallikelihood yields simple closed-form estimators when both μ and σ^2 are unknown: μ̂ = x̄ (the sample mean) and σ̂^2 = (1/n) ∑ (xi - x̄)^2 (the MLE for σ^2; using 1/(n-1) gives the unbiased estimator). If μ is known, σ̂^2 = (1/n) ∑ (xi - μ)^2.

Key properties include differentiability with respect to the parameters and a single peak in μ for fixed σ^2, enabling straightforward optimization. The normal likelihood underpins many statistical methods, including linear regression with normally distributed errors, hypothesis testing via likelihood ratio tests, and Bayesian inference where the likelihood combines with priors to form posterior distributions. Extensions cover multivariate normals and more complex hierarchical models.