I have two independent random normal variables $Y_1$ and $Y_2$. The sum is also normally distributed: $Y_1 + Y_2 \sim N(\mu_1+\mu_2,V_1+V_2)$
The normally-scaled inverse gamma (NSIG) distribution is a conjugate prior. I'd like to compute the posterior distribution for the sums $(\mu_1+\mu_2,V_1+V_2)$ which govern the distribution of $Y_1 + Y_2$, given a sample of $(Y_1,Y_2)$. What is this distribution? Is it still NSIG?