Parameters in an MVN

Assume MVN prior for mean, and Inverse Wishart for covariance. We get multivariate T posterior for mean, and Inverse Wishart posterior for covariance.
The above assumption shows how using the Inverse Wishart simplifies things because it is a conjugate prior

Assume and the data are complete.
Assume the likelihood and prior of the distribution of are normally distributed.
We have the following estimate for the posterior mean, which is normally distributed
Assume the likelihood and the prior of the covariance matrix are distributed based on the inverse Wishart distribution
We have the following estimate for the posterior covariance which is also inverse Wishart

The MAP estimate is given by
If we have a proper informative prior, we can rewrite the MAP estimate as

Where controls the shrinkage towards the prior
The prior covariance is determined using the MLE covariance. In obtaining We shrink off diagonal elements towards . This is shrinkage estimation or regularized estimation.
- One benefit of this is that the eigenvalues of the MAP estimate are closer to the true matrix rather than the MLE. However, this does not affect eigenvectors.

In the univariate posterior, we make use of the inverse-chi squared distribution as our posterior and prior distributions.
The posterior becomes
Assume the prior and likelihood follow a Normal Inverse Chi Squared distribution
The posterior is simply the Normal Inverse Chi-Squared distribution with updated parameters
The posterior marginal for follows the Inverse Chi Squared distribution
The posterior marginal for follows a Student T-distribution

If we assume an uninformative prior, we get a derivation for the Student t-test. We have

Where is the sample variance.
The test then constitutes computing the probability

We can simplify the posterior further using the t-statistic

And if , we have
Note: Here, we assume that is known, and parameter is unknown. The frequentist approach reverses the roles.

Assume the prior follows a Normal Inverse Wishartdistribution
Assume the likelihood is given by

Which is simply the product of the Gaussian and the Inverse-Wishart.
The posterior is simply the Normal-Inverse-Wishart distribution with updated parameters.
The posterior marginal for follows the Inverse Wishart distribution.
The posterior marginal for follows a multivariate Student T distribution
The posterior predictive is given by a Student-T distribution

The Library