Solved – Sample from a Normal-Inverse-Chi-Squared distribution

According to this document, on page 2, if $Z sim$ Normal Inverse $chi^2$ distribution$(m,k,r,s)$, then $Z = (W,V)$ such that $$dfrac{rs}{V} sim chi^2_r quad text{and} quad W|(V = v) sim Nleft(m, vk^{-1} right).$$

Thus to sample from a Normal-Inverse $chi^2$ distribution, you sample $V$ first and then sample $W$. For your parameters you have, $Z sim$ N-Inv-$chi^2$$(beta, sigma^2Omega^{-1}; v_0, s_0^2).$

Thus, you first sample $V = v$ such that, $$dfrac{v_0 s_0^2}{V} sim chi^2_{v_0}. $$

This can be done by first sample an $x$ from $chi^2_{v_0}$ and then set $v = v_0s_0^2/x$.

Next, using this $v$, you want to sample $W = w$ such that, $$W|(V = v) sim N(beta_0, vsigma^{-2}Omega). $$

Thus the draw $(w,v)$ will be a draw from the N-Inv $chi^2$.

(I am not familiar with Python, but I am sure there are functions that will help you draw from a $chi^2$ and Normal distributions.)