How Much Does One More Observation Move You?

Lesson 15 gave you the mechanics: posterior = Beta(α + k, β + n−k), addition of counts. This lesson asks you to build intuition for the shape of that update as data accumulates one point at a time — specifically, how much a single new observation is capable of moving an already-updated posterior.

Terms (standalone):

• Sharpening: as more data accumulates, a beta posterior’s variance shrinks — it becomes more concentrated (taller, narrower) around its mean. A posterior with α + β = 1000 is “sharper” (more confident, lower variance) than one with α + β = 4, even if both happen to have the same mean.
• Diminishing sensitivity: the more total weight (α + β, real + virtual trials) a posterior already carries, the less a single additional observation can move its mean. Going from 0 trials to 1 trial can swing a mean from “unknown” to “100% or 0%.” Going from 1,000 trials to 1,001 barely moves it at all — one more data point is now a tiny fraction of the total evidence.

Part 1 — Estimate (above): one more success on top of Beta(9, 5)

Give your best guess for the new posterior mean after adding one success to Beta(9, 5), plus a 90% interval you’re confident contains the true value. Think about it via the addition-of-counts rule before computing exactly — the point of this puzzle is to calibrate your intuition for how far one data point moves a mean that already represents 14 units of evidence.

Part 2 — Same single success, on a much younger posterior

Now imagine the same single new success, but observed against a much weaker starting point: Beta(1, 1) (the uniform prior, zero virtual trials — genuine ignorance). What’s the new posterior mean after that one success lands on Beta(1, 1)? Compute it exactly.

Part 3 — Compare the two movements

• Beta(9, 5) → one success → Beta(10, 5): mean goes from 9/14 ≈ 0.643 to 10/15 ≈ ?
• Beta(1, 1) → one success → Beta(2, 1): mean goes from 1/2 = 0.5 to 2/3 ≈ ?

Compute both new means exactly, then state each movement (new mean − old mean). Which posterior moved further from a single identical observation, and explain why in terms of total weight (α + β) before the new data point arrived.

Part 4 — General rule

Based on Parts 1–3, write a one-sentence rule for how much a single new observation can move a beta posterior’s mean, as a function of the posterior’s total weight (α + β) before that observation. (You don’t need a formula — a clear qualitative statement is enough, though if you want the exact form: the maximum possible movement from one observation is on the order of 1 / (α + β + 1).)