Q16. Interpreting identical interval endpoints
[4 marks]
A Bayesian analysis produces a 95% posterior credible interval [0.3,0.6] for a parameter \theta. A frequentist analysis happens to produce a 95% confidence interval with the same endpoints.
Explain the interpretation of each interval. Identify what is treated as random in each probability statement and explain why identical endpoints do not imply identical interpretations.
Definitions. C(X) denotes an interval computed from a random sample X; C({\boldsymbol{x}}) is its realised value. \mathbb P_\theta describes repeated sampling with \theta fixed. In contrast, \mathbb P(\,\cdot\mid{\boldsymbol{x}}) for the parameter is calculated from its posterior under the chosen model and prior. The stated 95% levels refer respectively to coverage and posterior probability.
For the Bayesian interval, \boxed{ \mathbb{P}(0.3\leq\theta\leq0.6\mid{\boldsymbol{x}})=0.95. } Conditional on the observed data, the assumed model, and the prior, there is 95% posterior probability that \theta lies between 0.3 and 0.6. The observed data are fixed, while uncertainty about \theta is represented by its posterior distribution.
For a frequentist 95% confidence-interval procedure C(X), \boxed{ \mathbb{P}_{\theta}\{\theta\in C(X)\}=0.95. } Here \theta is fixed and the interval is random because it is calculated from the random sample X. Under repeated sampling at the same true value of \theta, 95% of the intervals generated by the procedure would contain \theta. Once the data have been observed, the realised interval is fixed; the frequentist statement is not that this particular interval contains \theta with probability 0.95.
Therefore identical endpoints do not imply identical interpretations. The credible interval assigns posterior probability to values of \theta given the observed data, whereas the confidence interval describes the long-run coverage of a random interval-producing procedure.