 Solution

Write $v_i={\sigma }_i^2$ for each component variance and ${\tau }^2$ for the fixed prior variance denoted ${\sigma }^2$ in the source; these must not be confused. Let $\phi (x;\mu ,v)$ be the normal distribution probability density function and $D_j={\sum }_{i=1}^k{\omega }_i\phi (x_j;{\mu }_i,v_i)$. Independent observations and independent parameter priors give the posterior distributionIts support has $v_i>0$, ${\omega }_i>0$ and ${\sum }_i{\omega }_i=1$, with $\alpha ,\beta ,{\tau }^2,{ϵ}_i>0$. The component means have independent fixed-variance priors with a normal distribution, not means with variances proportional to $v_i$. The factor $e^{-\beta /v_i}$ follows from the stated inverse-gamma distribution. The later printed augmented expression instead has $e^{-\beta v_i}$ and an undefined prior mean; those are inconsistent with this specification. The intended model uses $e^{-\beta /v_i}$ and prior mean zero throughout.