Statistics
Maximum Likelihood
Which bell curve most likely produced this data? Let every point vote with the height of the bell above it, multiply the votes, and slide the bell until the product is as large as it can be. It lands on the sample mean and on the spread that divides by n.
datacandidate belleach point’s votelog-likelihood
Drag the bell (sideways moves it, up and down widens it) or drag a data point
Log-likelihood—
—
Chapter
sample mean x̄—
MLE σ̂, divides by n—
sample std s, divides by n − 1—
likelihood L (the product)—
What to observe
- A point far out in the tail gives a tiny vote, and one tiny factor drags the whole product toward zero. Slide the bell off to one side and watch the far points sink it.
- Products of many small numbers underflow, so we add logs instead: a small vote becomes a long cost bar. Same winner, easier numbers.
- Snap to best fit lands on μ̂ = x̄ and σ̂ = √(Σ(xᵢ − x̄)²/n). That σ̂ divides by n, not n − 1, so it comes out a little smaller than the sample standard deviation s.
- Too narrow and the outer points get almost no vote; too wide and nobody votes strongly. The best width balances the two.
- Drag one point far away in Explore: the best bell follows it and widens. Maximum likelihood for a bell listens to every point, outliers included.
Shortcuts:space run/pause · r reset · f fullscreen