Choosing a model and then reporting its conditional credible interval can discard a…

Choosing a model and then reporting its conditional credible interval can discard a substantial part of the posterior uncertainty. Quantify the effect in a small example. Suppose \[ \theta\sim0.8\,\delta_0+0.2\,N(0,1), \qquad Y\mid\theta\sim N(\theta,1), \] where \(\delta_0\) denotes a point mass at zero. You observe \(Y=3\). You may use the following full posterior without rederiving it: \[ \Pi(\,\cdot\mid Y=3) =(1-w)\delta_0+wN(m,s^2), \quad m=\frac32,\quad s^2=\frac12,\quad w=\frac{e^{9/4}/\sqrt2}{4+e^{9/4}/\sqrt2}. \] Let \(\Phi\) be the standard normal CDF and \(z=\Phi^{-1}(0.975)\). An analyst selects whichever posterior component has the larger probability. Identify the selected component, construct its conditional central \(95\%\) interval, and calculate that interval's probability under the **full** posterior. Next, let \(F\) be the CDF of the full posterior and define its left quantile by \[ Q(u)=\inf\{x:F(x)\ge u\},\qquad 0<u<1. \] Find the central quantile interval \([Q(0.025),Q(0.975)]\) and its actual full-posterior probability. Account explicitly for the atom at zero. Compare the two intervals, keeping posterior probability distinct from frequentist coverage.

First check which component is selected and whether its central interval includes zero.