Lesson 13 built the likelihood: for a fixed candidate rate p, how probable was the data you saw. But a likelihood alone isn’t a posterior — Bayes’ theorem (Lesson 6) needs a prior: your answer-sheet-of-plausibility over every candidate p before any data arrives. Multiply prior × likelihood, point by point across every candidate p, and (after rescaling so it sums to 1) you get the posterior.
Terms (standalone):
- Prior: a distribution over the unknown quantity (here, the true rate p) representing what you believe before seeing the data at hand. It’s still an answer sheet in the Lesson 12 sense — a plausibility value for every candidate p — just built from whatever you knew beforehand instead of from today’s data.
- Uniform prior: every candidate value gets equal plausibility. Often called an “uninformative” or “flat” prior — it encodes “I have no reason to favor any rate over any other,” not “I believe the rate is around 50%.” (A common beginner trap: uniform over [0,1] does not mean “I expect p ≈ 0.5” — it means every value from 0.001 to 0.999 starts out equally plausible.)
- Informative prior: concentrates plausibility on a narrower range, based on real prior knowledge — past data, domain expertise, physical constraints. The tighter and better-justified the prior, the more evidence it takes to move the posterior away from it.
- “Letting the data speak”: the folk phrase for using a weak/uniform prior so the posterior is driven almost entirely by the likelihood (the data) rather than by prior belief. It’s a choice, not a neutral default — choosing “no prior information” is itself an assumption, and it’s wrong exactly when you actually do have relevant prior information you’re discarding.
The puzzle (MCQ above)
Think about why an informative prior resists being moved: it isn’t magic, it’s math. A prior built from real data behaves, in the posterior-updating math you’ll formalize next lesson, almost as if it were itself a batch of “prior observations.” A prior distilled from 500 past buttons acts roughly like having already seen hundreds of data points before today’s 100 views even happened — so today’s 100 views, however real, are outvoted by the weight already baked into the prior.
Part 2 — When is “letting the data speak” (uniform prior) the right call?
Name a concrete situation where you’d genuinely want a uniform, uninformative prior over some unknown rate or quantity — where using an informative prior instead would actually be a mistake, not just a simplification.
Part 3 — When is an informative prior clearly correct, not just convenient?
Now name a situation (real or realistic) where ignoring strong prior knowledge — insisting on “let the data speak” with only a handful of new observations — would produce a genuinely bad, overconfident, or silly-looking estimate. What’s the general rule for when prior knowledge should dominate vs. defer to new data?
Part 4 — MCQ: what happens as data accumulates?
Suppose you keep collecting more views on that button — 100, then 1,000, then 100,000. What happens to the gap between the posterior under Prior A (uniform) and the posterior under Prior B (informative, centered at 2%), as the sample size grows?
(a) The gap stays exactly the same size no matter how much data arrives — priors are permanent. (b) The gap shrinks toward zero — with enough data, the likelihood eventually overwhelms any finite prior, and the two posteriors converge to (nearly) the same place. (c) The gap grows — more data makes priors matter more, not less. (d) The gap becomes undefined because posteriors are only valid for small sample sizes.