Option 1 is correct. “There’s a 95% probability that p lies between 0.61 and 0.74, given your
prior and the data” is exactly what a Bayesian credible interval claims — because a posterior
distribution literally is your (updated) probability distribution over p, and the credible
interval is just the range covering 95% of it. The probability statement is about p directly,
conditioned on what you believed going in and what you observed. That’s the whole point of treating
probability as degree of belief (Lesson 1) — it lets a sentence like this mean something.
Why option 2 is wrong here — but is the correct reading of the other kind of interval. Option 2 describes a frequentist confidence interval: a claim about a repeated procedure, not about this specific realized range or this specific parameter. It’s not wrong in general — it’s the textbook-correct interpretation of a confidence interval. It’s wrong here specifically because you didn’t build a confidence interval; you built a credible interval, which supports the stronger, more directly useful claim in option 1. This is the trap: swap the two intervals’ interpretations and you get a subtly, persistently wrong sentence that sounds almost identical to the correct one — 95% of misused statistics in the wild are exactly this substitution, made silently.
Why option 3 is wrong. That’s describing a predictive interval (Lesson 17’s territory) — a
range for a single new observation, not for the underlying parameter p. Different question
entirely; a credible interval for p says nothing directly about where any one future data point
will land (though the two are related — the predictive distribution is built from the posterior
over p).
Why option 4 is wrong. The interval says nothing about your confidence in the prior itself — it’s a downstream consequence of the prior plus the data, not a report card on the prior’s quality. (Checking whether a prior was reasonable is a separate exercise — model checking, sensitivity analysis — not something the credible interval tells you on its own.)
The one-sentence rule to keep: a credible interval is a probability statement about the parameter, conditional on your model; a confidence interval is a coverage statement about the procedure, averaged over hypothetical repetitions. If you ever catch yourself wanting to say “95% probability the true value is in here,” make sure you actually built a credible interval — because that sentence, applied to a confidence interval, is the single most common statistical misinterpretation in applied work.
Where this goes: the last stop before Stage 5’s computation tools is turning a posterior into an actual decision — not just a summary number or interval, but a choice about what to do, weighing what you stand to gain or lose.