Retrieval check answer. P(alarm) = 0.98×0.01 + 0.05×0.99 = 0.0098 + 0.0495 = 0.0593. P(defective | alarmed) = 0.0098 / 0.0593 ≈ 0.165 (about 16.5%) — most alarms are false alarms, the same base-rate story as Lesson 7, on a factory line instead of a clinic.
Part 1 — Virtual sample size: n₀ = 400/100 = 4. This prior is worth only 4 virtual
measurements — weak, compared to 16 real ones (4× as much real evidence as prior evidence). Expect the
posterior to land much closer to the data’s 180 ms than to the prior’s 200 ms.
Part 2 — Posterior mean: 184 ms.
prior precision = 1/100 = 0.01
data precision = 16/400 = 0.04
posterior precision = 0.01 + 0.04 = 0.05
posterior mean = (200 × 0.01 + 180 × 0.04) / 0.05
= (2 + 7.2) / 0.05
= 9.2 / 0.05
= 184
184 ms is 4× closer to the data (180) than to the prior (200) — distance 4 from the data, distance 16
from the prior, a 4:1 split, exactly matching the 4:1 ratio of data precision to prior precision
(0.04 : 0.01). This is the precision-weighted average doing exactly what its name says: whichever
side carries more precision (equivalently, more “weight” in the beta-binomial sense) pulls the
posterior more of the way toward itself.
Part 3 — With n = 100: posterior mean ≈ 180.77 ms.
data precision = 100/400 = 0.25
posterior precision = 0.01 + 0.25 = 0.26
posterior mean = (200×0.01 + 180×0.25) / 0.26 = (2 + 45)/0.26 = 47/0.26 ≈ 180.77
With 100 real measurements, data precision (0.25) dwarfs the prior’s fixed 0.01 — the posterior lands
almost exactly on the data’s 180, barely nudged by the prior at all. This is washing out the
prior (Lesson 14) in its precision-weighted form: any finite prior’s pull is a fixed, bounded
quantity, while data precision grows without bound as n grows, so enough real data eventually
swamps any reasonable prior — the same phenomenon, in a completely different distribution family.
Why this is conjugacy, concretely. Neither computation touched an integral. You added two
precisions and took a weighted average of two means, and the algebra of normal distributions
guarantees the result is itself normal. That’s what “conjugate” buys you: a prior family paired
with a matching likelihood family such that the posterior stays in the same family, collapsing the
update to arithmetic on that family’s own parameters. Beta-binomial adds counts (α, β);
normal-normal adds precisions and takes a weighted mean. Different parameters, same underlying
shape: combine what you believed with what you saw, weighted by how much each side is worth trusting.
Where this goes: conjugate pairs like these two are the convenient exception, not the rule — most real priors and likelihoods you’d actually want to combine don’t have a tidy closed form at all. Next lesson previews exactly what breaks conjugacy, and what you do once the closed form runs out.