AIDAID202 · Lectures 7–8

Lecture 7 · Maximum likelihood

Coin MLE: likelihood you can see

MLE chooses the parameter value that makes the observed data most probable. For a Bernoulli coin, that turns out to be the observed frequency of heads.

Predict first: You observe H,H,T,H,T. Without using the formula, where should the likelihood curve peak?

Observed flips

Likelihood (scaled)MLETrue θ
Heads / n—
θ̂MLE—

How stable is the estimate?

Why does the frequency appear?

For H heads and T tails, the likelihood is proportional to θ^H(1−θ)^T. Maximizing its log gives H/θ − T/(1−θ)=0, so θ̂ = H/(H+T).

As the number of flips grows, repeated estimates concentrate around the true parameter.