A covariance matrix, a banded matrix, and a fast transform

Abstract: 

It often happens that we know some but not all of the numbers in a covariance matrix. How should we choose the missing entries to complete the matrix? Dempster suggested completing to maximize the determinant. 

Terry Speed recognized a neat "local inverse formula" in that completion. That matrix completion involves some attractive linear algebra that I will highlight. There is also a nice connection to banded matrices with banded inverses, and from there, to fast transforms.