A Gaussian is completely described by its mean and covariance. The level sets of its density are the ellipses x⊤Σ−1x = c, whose axes are the eigenvectors of Σ and whose semi-axes are √(cλi). Because that quadratic form is χ2 distributed, a specific c makes the ellipse a confidence region: in two dimensions c = −2 log(1 − p), so 90% always sits at c = 4.61. Draw samples and count them to check that the promise is kept.
| ellipse | should hold | actually holds |
|---|---|---|
| 50% | 50.0% | – |
| 90% | 90.0% | – |
| 99% | 99.0% | – |
Drag the black probe point: what matters is not how far it is in the ordinary sense, but how far in the geometry Σ defines.