First, a quick retrieval check — different concept, new setting.
A factory sensor should catch defective units. Historically, 1% of units are defective (base rate). The sensor alarms on 98% of actual defects, but also false-alarms on 5% of good units. It alarms on today’s unit. What’s P(actually defective | alarmed)? Same machinery as Lesson 6, a different noun than the diagnosis it was first taught with.
(Worked out in the solution.)
Every update in Stage 2 used the same machinery: beta prior, binomial-flavored data, add counts. That pairing — a prior family that, combined with a specific likelihood, produces a posterior in the same family — is called conjugacy. Beta-binomial is one conjugate pair. It isn’t the only one.
A second conjugate pair: normal-normal. When your prior belief about an unknown mean is itself a normal distribution, and your data is normally distributed with a known variance, the posterior is also normal, with a closed form — no integration required, exactly like beta-binomial.
The closed form, in terms of precision (precision = 1 / variance; a bigger number means a
tighter, more confident distribution):
posterior precision = prior precision + data precision
posterior mean = (prior_mean × prior_precision + sample_mean × data_precision) / posterior precision
where data precision = n / σ² (n new observations, each with known per-observation variance σ²).
It’s a precision-weighted average of the prior mean and the sample mean — the same spirit as
beta-binomial’s “more weight resists being moved,” just phrased in variance instead of counts.
Your scenario. You’re estimating the true average response time of an API endpoint.
- Prior belief: mean = 200 ms, variance = 100 (so prior precision =
1/100 = 0.01). - New data:
n = 16fresh measurements, sample mean = 180 ms. Individual measurements are known (from long operational history) to have varianceσ² = 400— so data precision =n / σ² = 16 / 400 = 0.04.
Part 1 — Virtual sample size. Beta-binomial’s prior “weight” was α + β virtual trials, directly
comparable to real trials. Normal-normal has an analogous idea: a prior with variance σ0² behaves
like it’s worth n₀ = σ² / σ0² virtual observations of the same per-observation noise as the real
data. Compute n₀ for this prior. Is 16 real measurements a lot or a little, compared to that?
Part 2 — Compute the posterior mean (the numeric answer above). Use the precision formula above. Should the posterior land closer to 200 (the prior) or 180 (the data), and by how much, given the two precisions involved?
Part 3 — More data. Suppose instead you’d collected n = 100 measurements at the same sample mean
of 180 ms. Recompute the posterior mean. Which way does it move relative to Part 2’s answer, and why —
in terms of the same washing-out-the-prior idea from Lesson 14?