Exercise 2: Confidence Interval vs. Credible Interval Are Not the Same Claim — Possible Solution ==================================================================================== WHAT'S WRONG WITH THE COLLEAGUE'S CLAIM ------------------------------ The colleague is treating a confidence interval and a credible interval as two different computation paths arriving at the same underlying meaning - but per this chapter's own frequentist-vs- Bayesian distinction, they make genuinely different KINDS of claims, not just different arithmetic for the same claim. WHAT A 90% CONFIDENCE INTERVAL ACTUALLY MEANS ------------------------------ Per Chapter 3's own frequentist definition, a 90% confidence interval means: if this exact sampling-and-interval-building procedure were repeated many times, about 90% of the resulting intervals would contain the true (fixed, unknown) population parameter. It says nothing about the probability that THIS particular, already-computed interval contains the true value - that value either is or isn't in the interval, with no probability left to assign to it once it's been computed. WHAT A 90% CREDIBLE INTERVAL ACTUALLY MEANS ------------------------------ Per this chapter's own Bayesian framework, a 90% credible interval means exactly what people usually WANT a confidence interval to mean: there is a genuine 90% probability that the true parameter value falls within this specific interval, given the prior and the observed evidence. This is only a coherent statement in the Bayesian framework, where the parameter itself is treated as having its own probability distribution rather than being a fixed constant. WHY THIS DISTINCTION MATTERS, NOT JUST AS A TECHNICALITY ------------------------------ The two intervals are built from different underlying frameworks (fixed-parameter vs. distributed-parameter) and, in general, are not guaranteed to give even numerically identical bounds for the same data - they can differ, especially when a strong prior is involved. Treating them as interchangeable risks stating a Bayesian-flavored claim ("90% probability the truth is in here") while having actually only computed the frequentist version, which per Chapter 3's own warning does not support that claim at all. WHY THIS WORKS AS AN ANSWER ------------------------------ The explanation is grounded directly in this chapter's own stated definitions of both interval types, explicitly cites Chapter 3's own confidence-interval meaning to contrast against it, and explains why conflating the two isn't a harmless simplification but a genuine misstatement of what has actually been established.