 Gaussian practical-null mixture

A Gaussian practical-null mixture compares a narrow centered normal distribution prior with a wider centered normal distribution prior. Neither component is a point mass. With observed mean $y\mid \beta \sim N(\beta ,1/n)$ and prior precisions $q_0>q_1>0$, its model predictive variances are $V_i=1/n+1/q_i$, and the Bayes factor isThe overall posterior density is a Bayesian model averaging mixture with component means $ny/(n+q_i)$. A wide-prior penalty can favor the practical null near zero; its dominance depends quantitatively on both prior scales and the observation, not merely on the observation being of order $n^{-1/2}$.