MLE для смещенной монеты
MLE для смещенной монеты
Ответить самому
Сначала сформулируйте ответ как на собеседовании, затем откройте разбор и оцените себя.
Короткий ответ
The MLE is p_hat = k / n.
Полный разбор
The likelihood of observing k heads and n-k tails is proportional to:
L(p) = p^k (1-p)^(n-k).
The log-likelihood is:
l(p) = k log p + (n-k) log(1-p).
Differentiate and set to zero:
k / p - (n-k) / (1-p) = 0.
Rearrange:
k(1-p) = (n-k)p, so k = np, and p_hat = k / n.
The second derivative is negative for p in (0, 1):
-k / p^2 - (n-k) / (1-p)^2 < 0.
So the stationary point is a maximum. Boundary cases are intuitive: if k=0, p_hat=0; if k=n, p_hat=1.
Теория
For i.i.d. Bernoulli observations, the maximum likelihood estimate is the sample mean.
Типичные ошибки
- Differentiate the likelihood directly and lose terms.
- Forget to check that the stationary point is a maximum.
- Miss the boundary cases k=0 and k=n.
Как отвечать на собеседовании
- Use log-likelihood; it turns products into sums and keeps the derivation clean.
- Mention the second derivative or concavity to close the derivation.