Lessons 15 and 19 gave you two conjugate pairs: beta prior + binomial likelihood, and normal prior + normal likelihood (known variance). Both let you skip integration entirely — you just update a couple of parameters with arithmetic, and the posterior is guaranteed to land in the same family you started in.
It’s tempting to conclude “as long as the data is Bernoulli/binomial, I get a closed form.” That’s not quite the rule. Conjugacy is a property of the pair — a specific prior family matched to a specific likelihood family — not a property of the likelihood alone. If your actual prior belief doesn’t genuinely belong to the matching family (it just happens to have been approximated by one for convenience), the guarantee doesn’t transfer to what you actually believed.
Four setups below. In three of them, the prior you’re using really is the distribution you believe, and it’s from the family that’s conjugate to the likelihood — so the update is exact arithmetic, closed form, done. In one of them, the “prior” being fed into the update is a convenient stand-in for a belief that doesn’t actually have that shape (it’s bimodal — two separate humps of plausibility, maybe because you think the true rate is either “low” or “high” but probably not in between). Using the closed-form update on that one silently answers a slightly different question than the one you actually meant to ask.
Which one is it?