If \((x,y)\) has a Normal-ExpGamma distribution with parameters \(\mu\), \(\kappa\),
\(\alpha\), 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 normal distribution with expectation \(\mu\) and logged standard deviation
\(\kappa + y\). The probability density is proportional to
$$
f(x,y)=\exp(-(2\alpha + 1)y - e^{-2y}(\beta + e^{-2\kappa}(x-\mu)^2/2))
$$