Confidence Intervals and Decision-Theoretic Inference
Intervals as repeated-sampling procedures
Frequentist confidence intervals
A confidence interval is a random interval procedure that covers the fixed parameter of the time in repeated samples under the assumed model.
Coverage matters
A confidence interval is evaluated by its long-run coverage probability, not by the probability that the realized interval contains the parameter.
What is the correct frequentist interpretation of a confidence interval?
Coverage is a property of the interval-generating procedure.
Correct answer: The procedure covers the true parameter 95% of the time
What is the key ingredient of a confidence interval derived from an estimator?
Typical form is .
Correct answer: An estimate plus a measure of uncertainty, often a standard error and quantile.
Decision theory
Decision theory formalizes inference by choosing actions to minimize expected loss or posterior expected loss. Loss functions encode the relative cost of estimation errors.
Common losses
Under squared error loss, what estimator minimizes posterior expected loss?
The posterior mean minimizes posterior expected squared error loss.
Correct answer: Posterior mean
Why are confidence intervals often preferred over point estimates alone?
Intervals provide a range of plausible values and improve scientific interpretation.
Correct answer: They express the uncertainty and sampling variability of the estimate.