The Normal-Inverse-Gamma distribution is often written as
$N(\phi | \mu, \sigma^2 \Sigma) IG(\sigma^2 | \alpha, \beta),$
and used as a conjugate prior for a linear model given observations
$y_t \sim N(y_t | x_t \phi, \sigma^2).$
However, the model would still be conjugate even if the prior would be written as
$N(\phi | \mu, \Sigma) IG(\sigma^2 | \alpha, \beta),$
which begs the question of why is the variance of the Normal part of the prior scaled by $\sigma^2$ in the first place, as this makes some derivations longer and more problematic. Most importantly, is there some clear benefit regarding the quality of the inferences?