  Freedman's inequality is a martingale counterpart to Bernstein's inequality.
This result shows that the large-deviation behavior of a martingale is
controlled by the predictable quadratic variation and a uniform upper bound for
the martingale difference sequence. Oliveira has recently established a natural
extension of Freedman's inequality that provides tail bounds for the maximum
singular value of a matrix-valued martingale. This note describes a different
proof of the matrix Freedman inequality that depends on a deep theorem of Lieb
from matrix analysis. This argument delivers sharp constants in the matrix
Freedman inequality, and it also yields tail bounds for other types of matrix
martingales. The new techniques are adapted from recent work by the present
author.
