Likelihood

165. Likelihood

ℒ︀ (𝜃 | 𝑥) = 𝑓 (𝑥 | 𝜃)

Where:

Same formula, read two ways — what is held fixed and what varies swap:

𝜙 (𝑥; 𝜇, 𝜎) = \frac{1}{𝜎 \sqrt{2 𝜋}} exp (- \frac{{(𝑥 - 𝜇)}^{2}}{2 𝜎^{2}})

PDF: parameters fixed, sweep $𝑥$ — a curve over outcomes that integrates to 1.
Likelihood: data fixed, sweep the parameters — not a density in $𝜇, 𝜎$ , so it need not integrate to 1.

For $𝑛$ independent observations the likelihood is the product of per-point densities:

ℒ︀ (𝜇, 𝜎 | 𝑥_{1}, \dots, 𝑥_{𝑛}) = \prod_{𝑖 = 1}^{𝑛} 𝜙 (𝑥_{𝑖}; 𝜇, 𝜎)