Likelihood Surface Explorer (2D) — Maximum Likelihood Estimation

Statistical inference means drawing conclusions about an unknown population from a finite sample, and maximum likelihood is the workhorse method for doing it: pick the parameter values that make the observed data most probable. This 2D counterpart draws a random sample from a Normal(μ, σ) population, computes its full log-likelihood function on a real (μ, σ) grid, and renders it as a heatmap with contour rings. The maximum-likelihood estimate sits exactly at the peak found by grid search, and a 95% confidence ellipse — built from the numerically-computed Hessian of the log-likelihood at that peak — shows exactly how the curvature of the surface encodes estimation uncertainty. Smaller samples produce a flatter surface and a larger ellipse; larger samples sharpen the peak and shrink it, which is the same Fisher-information argument behind the standard error reported next to almost every fitted statistic in science.