What is a reasonable normal density to associate with some data?
Suppose $x^1, \dots , x^n$ be a dataset in
$\R ^d$.
The empirical mean of
the dataset is the real number
\[
\frac{1}{n} \sum_{k = 1}^{n} x^k.
\] \[
\frac{1}{n} \sum_{i = 1}^{n} (x^k - \frac{1}{n} \sum_{k =
1}^{n} x^k) (x^k - \frac{1}{n} \sum_{k = 1}^{n} x^k)^\top .
\]