# Why model the variance-covariance matrix as an inverse-Wishart distribution in bayesian portfolio analysis?

I am following Risk and asset allocation (Attilio Meucci,2007). I must say I am enjoying this reading quite a lot so I hope nobody takes my question as a critique on the text.

When we are introduced to the Bayesian approach in asset-allocation we realise we need to specify a joint distribution for the mean and variance of the data (respectively $\mu, \Sigma$) to obtain a prior we can plug in the Bayes formula. We recall that $f(\mu, \Sigma) = f(\mu | \Sigma)f(\Sigma)$.

So For the conditional mean $\mu|\Sigma$ the text chooses a normal:

$$\mu|\Sigma \sim N\Bigg(\mu_0 , \frac{\Sigma}{T_0}\Bigg)$$

This seem like a good choice that is also pretty intuitive.

Instead the text models the variance ,$\Sigma$, as an inverse Wishart distribution:

$$\Omega = \Sigma^{-1} \sim W\Bigg( v_0, \frac{\Sigma^{-1}}{v_0} \Bigg)$$

The text then goes on to give some explanations but I was wondering: are there any intuitive reasons why this is a good way to model variance?