Posterior credibility does not ensure nominal frequentist coverage

The joint density/mass is $1\cdot \theta$ at $Y=1$ and $1\cdot (1-\theta )$ at $Y=0$. Its marginal masses are ${\int }_0^1\theta d\theta =1\mathrm{/}2$ and ${\int }_0^1(1-\theta )d\theta =1\mathrm{/}2$. Therefore the posterior densities are $p(\theta \mid Y=1)=2\theta$ and $p(\theta \mid Y=0)=2(1-\theta )$; each conditional law integrates to one and satisfies the defining conditioning identity.

Their CDFs on $[0,1]$ are $F_1(x)=x^2$ and $F_0(x)=1-(1-x{)}^2$. Hence the central 95% intervals are $C(1)=[\sqrt{0.025},\sqrt{0.975}]$ and $C(0)=[1-\sqrt{0.975},1-\sqrt{0.025}]$. Each has posterior mass $0.975-0.025=0.95$, also in accordance with [F1, F2].

For fixed true ${\theta }_0=0.005$, $\sqrt{0.025}>0.005$ because $0.025>{0.005}^2$. Also $1-\sqrt{0.975}>0.005$ because $0.975<{0.995}^2$. Thus ${\theta }_0$ lies below the lower endpoint of both possible intervals, so $P_{{\theta }_0}\{{\theta }_0\in C(Y)\}=0$, not $0.95$. This verifies that posterior credibility is not a frequentist coverage guarantee.