Likelihood Surface Explorer — Maximum Likelihood Estimation

Sample

Sample size n 40

True mean μ 0.0

True std σ 1.5

Probe the surface

Guess μ̂ 0.0

MLE peak (μ̂, σ̂)
Your probe (guess μ, σ̂)

Live readouts

MLE μ̂

0.00

MLE σ̂

0.00

Log-likelihood drop at probe (peak − probe)

0.00

Asymptotic SE(μ̂) = σ̂ / √n

0.00

How it works

Given samples x₁…xₙ assumed drawn from a Normal(μ, σ) distribution, the log-likelihood as a function of the two unknown parameters is:

ln L(μ,σ) = -n/2·ln(2π) - n·ln(σ) - (1 / (2σ²))·Σ(xᵢ - μ)²

The surface you see plots exactly this function over a grid of (μ, σ) candidates, height and color both encoding ln L. The maximum-likelihood estimate is simply the (μ,σ) pair that sits at the highest point:

μ̂ = (1/n)Σxᵢ            (sample mean)
σ̂ = √[(1/n)Σ(xᵢ - μ̂)²]  (biased MLE std, divides by n not n-1)

n / true μ / true σ — control the population the data is actually drawn from, then redraw a fresh sample from it.
Guess μ̂ slider — slides a probe along the ridge σ = σ̂, showing how ln L falls away as your guess moves off the true MLE. The drop is quadratic near the peak — that curvature is the Fisher information.
Fisher information I(μ) = n/σ² measures exactly this curvature. Its inverse square root gives the asymptotic standard error SE(μ̂) = σ̂/√n: a narrow, sharply-peaked ridge means a small n or large σ can't pin μ down precisely, while a large n makes the surface steep and the estimate tight — try dragging n up and watching the ridge narrow.

This is the mechanism behind virtually every "best-fit" number reported in science: regression coefficients, particle masses, epidemiological rates — all are MLEs, and their reported uncertainty almost always comes from this same curvature argument.

MLE peak Probe

⚠ Couldn't load the 3D engineThree.js failed to load from the CDN. Check your connection and reload.