Part 1 — Exact answer: 0.667
Beta(9, 5) plus one success (k=1, n−k=0) → Beta(10, 5). Mean = 10 / 15 = 0.6667.
If your estimate landed close to 0.643 (barely moved from Lesson 15’s answer) with a tight interval, your intuition for “diminishing sensitivity” is already well-calibrated — a posterior carrying 14 units of weight barely notices one more data point. If your interval was wide or your guess far off, that’s exactly the intuition this lesson is building.
Part 2 — Same success on Beta(1, 1)
Beta(1, 1) plus one success → Beta(2, 1). Mean = 2 / 3 = 0.6667.
(Yes — coincidentally the same numeric value as Part 1’s answer. That coincidence is a trap if you stop here; Part 3 shows why comparing the endpoints alone is the wrong comparison.)
Part 3 — The movements, compared
- Beta(9,5) → Beta(10,5): 9/14 (≈0.6429) → 10/15 (≈0.6667). Movement ≈ 0.024.
- Beta(1,1) → Beta(2,1): 1/2 (0.5) → 2/3 (≈0.6667). Movement ≈ 0.167.
Same single new data point (one success), landing on two different posteriors — and the weak posterior moved about 7× further than the strong one, even though (by coincidence of the specific numbers chosen) both happened to arrive at the same final mean.
Why: before the new observation, Beta(9,5) already carried α+β = 14 units of weight (prior + data so far); Beta(1,1) carried only α+β = 2. Adding one success is adding 1 unit of weight to a pool of 14 (a ~7% change in composition) versus adding 1 unit to a pool of 2 (a 50% change in composition). The fraction of total evidence that the new point represents is what determines how far the mean can move — not anything about the new observation itself, which was identical in both cases.
Part 4 — The general rule
A single new observation can move a beta posterior’s mean by roughly 1 / (α + β + 1) — i.e.,
inversely proportional to how much total weight (real + virtual trials) the posterior already
carries. Check it against Part 3: Beta(9,5) had α+β=14, so max movement ≈ 1/15 ≈ 0.067 (our actual
movement of 0.024 falls within that ballpark, depending on direction); Beta(1,1) had α+β=2, so max
movement ≈ 1/3 ≈ 0.33 (our actual movement of 0.167 is again in that ballpark). The exact bound
depends on which direction the observation pushes, but the scaling — inversely with existing
weight — is the reliable takeaway.
This is exactly the mechanism behind sharpening: as α+β grows, the posterior doesn’t just center somewhere, it becomes more resistant to being knocked around by any one new data point, i.e. its variance shrinks. A posterior’s “confidence” isn’t a separate quantity you compute alongside the mean — it’s a direct, mechanical consequence of how much weight has accumulated in α+β.
The pattern
| Posterior before | α+β (weight) | +1 success | New mean | Movement |
|---|---|---|---|---|
| Beta(1,1) | 2 | Beta(2,1) | 0.667 | 0.167 |
| Beta(9,5) | 14 | Beta(10,5) | 0.667 | 0.024 |
| Beta(99,50) | 149 | Beta(100,50) | 0.667 | ≈0.002 |
Rule: the same single observation moves a young, low-weight posterior a lot and a mature,
high-weight posterior barely at all — one more data point matters in proportion to how small a
fraction of total accumulated evidence it represents, roughly 1/(α+β+1).
Where this goes: Stage 2 closes with one more idea — the posterior predictive: given everything the posterior now believes about p, what do you actually expect to see on the next single trial? Stage 3 then formalizes “yesterday’s posterior is today’s prior” — the sequential updating you’ve been doing by hand all through Stage 2, made explicit and general.