If \((x,y)\) has a multivariate
Normal-ExpGamma distribution with parameters \(\mu\), \(P\),
\(\alpha\), and \(\beta\), then the marginal distribution of \(y\) has an ExpGamma
distribution with parameters \(\alpha\), \(\beta\), and -2, and conditionally on \(y\),
\(x\) has a multivariate
normal distribution with expectation \(\mu\) and precision matrix
\(e^{-2y}P\). The probability density is proportional to
$$
f(x,y)=\exp(-(2\alpha + k)y - e^{-2y}(\beta + (x-\mu)^tP(x-\mu)/2))
$$
where \(k\) is the dimension.