 Solution

Integrating the positive component's normal distribution prior gives its prior predictive distribution:The negative component gives $Y_i\mid -\sim N(0,1)$. Let $f_1(y)={\varphi }_{1+V}(y)$ and $f_0(y)={\varphi }_1(y)$ denote those centred normal densities. The point-null mixture prior assigns component odds $q/(1-q)$, so the posterior odds of the positive component areIts posterior probability is $O_i/(1+O_i)$. Evidence depends on $y_i^2$, so large effects of either sign favour the positive component. This is the marginal distribution before observing $y_i$, not the posterior predictive law for a second observation conditional on it.