# How is the sum of normally-scaled inverse gamma variates distributed?

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?

Given $y_1$ and $y_2$, the posteriors on $(\mu_i,V_i)$ are independent: $$(\mu_i,V_i)|y_i \sim \mathcal{N}(\xi_i,V_i/\lambda^\mu_i)\times \mathcal{IG}(\lambda^{\sigma}_i,\alpha_i)\,,$$ where the normal distribution is the conditional distribution of $\mu_i$ given $y_i$ and $V_i$ [see formula (2.7) in Bayesian Core!] Then this means that, conditional on $(V_1,V_2)$ and $(y_1,y_2)$, $$\mu_1+\mu_2\sim \mathcal{N}(\xi_1+\xi_2,V_1/\lambda^\mu_1+V_2/\lambda^\mu_2)\,.$$ Unfortunately, the marginal distribution of $V_1+V_2$ is not so straightforward since it is the sum of two inverse gamma random variables with different scale and shape parameters... Even the sum $V_1^{-1}+V_2^{-1}$ does not enjoy a simple expression when the scale parameters are not the same!
• Note also that the posterior of $\mu_1+\mu_2$ does not depend on $V_1+V_2$... – Xi'an Feb 1 '12 at 21:33
• @user388027: given that you know (in closed form) the posterior distribution of $(V_1,V_2)$, you can use simulation to learn about everything on the posterior distribution of $V_1+V_2$. – Xi'an Feb 1 '12 at 21:37