Q8. Equal-tailed and highest-density intervals
Suppose a continuous parameter \theta has posterior density p(\theta\mid{\boldsymbol{x}})=2\theta, \qquad 0<\theta<1, and zero elsewhere.
(a) Find a 75% equal-tailed posterior credible interval. [3 marks]
(b) Find a minimum-width 75% posterior credible interval. Sketch the posterior density and explain why your interval is a highest-density interval. Exact expressions are sufficient. [3 marks]
(c) A frequentist confidence-interval procedure happens to return the same endpoints as one of your intervals. Explain the difference between a 75% confidence statement and a 75% posterior credible statement. [2 marks]
Definitions. For a continuous posterior, F(u)=\mathbb P(\theta\le u\mid{\boldsymbol{x}}) and F^{-1} is its quantile function. An equal-tailed interval puts half the omitted probability in each tail. A highest-density interval (HDI, also called HPD) includes points of at least as high density as those excluded. Here the posterior density is 2\theta on (0,1) and the required probability is 0.75.
For 0<\theta<1, the posterior distribution function is F(\theta)=\int_0^\theta 2u\,du=\theta^2, \qquad F^{-1}(p)=\sqrt{p}.
(a) Equal-tailed 75% credible interval
An equal-tailed 75% interval leaves probability 0.125 in each tail. Therefore its endpoints are F^{-1}(0.125)=\sqrt{\frac18}, \qquad F^{-1}(0.875)=\sqrt{\frac78}. Thus \boxed{ \left[\sqrt{\frac18},\sqrt{\frac78}\right] } is the 75% equal-tailed posterior credible interval.
(b) Minimum-width 75% credible interval
The posterior density 2\theta increases throughout (0,1), so the highest-density interval must contain the largest values of \theta and end at 1. Let it be [c,1]. Then \mathbb{P}(c\leq\theta\leq1\mid{\boldsymbol{x}}) =1-F(c)=1-c^2=0.75, which gives c=1/2. Hence the minimum-width interval is \boxed{\left[\frac12,1\right]}.
Every included point has density at least as large as every excluded point, so [1/2,1] is a highest-density interval. Equivalently, among intervals [a,b] satisfying b^2-a^2=0.75, its width (b-a)=0.75/(a+b) is minimised by taking the largest possible a+b, namely b=1 and a=1/2.
(c) Confidence versus credibility
A 75% confidence procedure has repeated-sampling coverage \mathbb{P}_{\theta}\{L(X)\leq\theta\leq U(X)\}=0.75. The probability refers to the random interval procedure; after observing the data, the realised frequentist interval is fixed.
A 75% credible interval [a,b] satisfies \mathbb{P}(a\leq\theta\leq b\mid{\boldsymbol{x}})=0.75. This probability refers directly to the posterior distribution of \theta given the observed data, prior, and model. Identical numerical endpoints do not change these different interpretations.