= Gaussian practical-null mixture
{c}
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> is
$$
B_{01}(y)=\sqrt{V_1/V_0}\exp[-y^2(V_0^{-1}-V_1^{-1})/2].
$$
The 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 $n^{-1/2}$.
Back to article page
