Every update so far has 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’s not 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 mean and variance. No integration required, just like beta-binomial.
The closed form, expressed in 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 observations, each with known per-observation variance σ²).
Notice the shape: it’s a weighted average of the prior mean and the sample mean, weighted by how
confident each one is — exactly the same spirit as “more data pulls you further,” just phrased in
variance instead of beta’s counts.
Your scenario. You’re estimating the true average session length on a website.
- Prior belief: mean = 5.0 minutes, variance = 1.0 (so prior precision = 1/1.0 = 1).
- New data:
n = 9sessions observed, sample mean = 6.5 minutes. Individual sessions are known (from long historical experience) to have varianceσ² = 4— so the sample mean’s variance isσ²/n = 4/9 ≈ 0.444, giving data precisionn/σ² = 9/4 = 2.25.
Before computing anything, use the precision-weighted-average intuition: the data’s precision (2.25) outweighs the prior’s precision (1), so the posterior mean should land closer to the sample mean (6.5) than to the prior mean (5.0) — roughly how much closer is exactly what you’re estimating.
Give your best guess for the posterior mean (in minutes), plus a 90% interval you’re confident contains the true value, before you compute it exactly.