# 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?

## 1 Answer

If you give a covariance matrix an inverse Wishart prior, then it simplifies a lot of math in the calculations. This is called a conjugate prior. If you don't understand conjugate priors, you might want to work through the math on the univariate normal case with an inverse gamma or chi square prior for the variance. The Wishart distribution is just a generalization of chi square or gamma to multivariate. Since you use inverse gamma or chi square for variance, you do the inverse of the covariance matrix when generalizing to multivariate.