 Solution

Use the data augmentation labels to separate the mixture into groups. Define $n_i={\sum }_j{1}_{\{z_j=i\}}$, $s_i={\sum }_{z_j=i}{x}_j$ and $Q_i({\mu }_i)={\sum }_{z_j=i}(x_j-{\mu }_i{)}^2$. The intended joint posterior distribution is proportional toSumming over all allocation labels recovers the marginal mixture posterior. The Gaussian mixture Gibbs updates with independent priors follow by collecting the terms in each parameter.

For ${\mu }_i$, the conditional exponent isCompleting the square givesFor $v_i$, the conditional density is proportional to $v_i^{-(\alpha +n_i/2)-1}\exp [-(\beta +Q_i({\mu }_i)/2)/v_i]$, henceThe independent prior on ${\mu }_i$ adds neither a $v_i^{-1/2}$ factor nor a ${\mu }_i^2/v_i$ penalty. Finally, combining the allocation counts with the Dirichlet distribution prior givesA Gibbs sampler successively draws from these conditionals, using the most recently updated values. An empty component has $n_i=s_i=Q_i=0$, so its mean and variance conditionals reduce to their proper priors without dividing by an undefined group sample mean.