Bayesian conjugate priors
Watch a prior become a posterior. Pick a conjugate pair, set your prior belief and your data on sliders, and see the posterior slide from one towards the other — with the exact credibility weight Z doing the pulling.
Posterior mean 0.529
The posterior mean is a credibility blend of the data and the prior:
0.529 = 0.59 × 0.700 (data mean) + 0.41 × 0.286 (prior mean)
Z = n / (n + α + β) = 0.59. Slide the data up and Z climbs towards 1: evidence overwhelms opinion. A sharper prior (bigger α + β, or bigger λ) resists longer. This is exactly the credibility premium Z·X̄ + (1 − Z)·μ, which is why the Bayesian chapter and the credibility chapter are the same chapter wearing different notation.
Why conjugacy matters
Bayes’ theorem says posterior ∝ likelihood × prior, which in general needs an integral nobody wants to do in an exam. Conjugate pairs are the families where the algebra closes: a Beta prior with Binomial data gives a Beta posterior, a Gamma prior with Poisson counts gives a Gamma posterior, and updating collapses to adding the data to the parameters. That is the entire trick behind most CS1 Bayesian questions — spot the pair, add the counts, read off the posterior mean under quadratic loss. The decomposition at the bottom of the tool is the deeper point: every conjugate posterior mean is a credibility formula, blending the data mean and the prior mean with a weight that grows with evidence.
Keep going. The same blend runs the credibility theory calculator, and both distributions live on the distributions cheat sheet. Memori is a flashcard app built by actuarial students, with a ready-made CS1 set in the shop. Join the beta.
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