Suppose $$x|\mu_1,\mu_2\sim N(\mu_1+\mu_2,1),$$ $$\mu_i\sim N(0,\sigma^2),\ iid$$ $$\sigma^2\sim Inv-Gamma(a,b).$$
and assume the first observed $x=1.$ We want to use Gibbs to sample $x.$
I already got the closed forms of full conditional distributions: $$f(x|\mu_1,\mu_2, \sigma^2),f(\mu_1|x,\mu_2, \sigma^2),f(\mu_2|x,\mu_1, \sigma^2),f(\sigma^2|x, \mu_1,\mu_2).$$ But the problem is what $$f(\mu_1,\mu_2, \sigma^2|x)$$ is？ Since we should use $f(\mu_1,\mu_2, \sigma^2|x=1)$ to generate the initial values of $\mu_1,\mu_2, \sigma^2.$ It seems a Normal-Inverse Gamma distribution which is not easy to be simplified?