Henry Baker wrote: ----- Pick 2 random vectors in 4D. What is the angle between them, in maximum likelihood ? ----- The usual definition of "maximum likelihood" is the parameter with the maximum probability. But for any given angle, its probability is 0. Also, since angle between two vectors depends only on their directions, let's assume we're considering only unit vectors. Then all such vectors form a unit 3-sphere S^3. We may as well fix one of the vectors, say as u = (1,0,0,0) in S^3. The vectors v forming an angle t with u form the set A(t) = {v ∊ S^3 | <u,v> = cos(t)} where <u,v> is dot product. It's easy to see that A(t) is a 2-dimensional sphere of radius = sin(t), and so its 2-volume is vol_2(A(t)) = 4 π sin(t)^3/3. It makes sense to interpret the "maximum likelihood" angle as that angle maximizing this volume. Clearly that occurs when sin(t) = 1 and t = π/2. —Dan