Volumes of Bitnets http://omega.albany.edu:8008/bitnets Carlos Rodriguez SUNY Albany.

Volumes of

Meet first bit Adam 1

Meet second bit 2 1 or 2 1

Complete Bitnets Simplex P Cube T Sphere S t p t p t p x x x 3 3 2 p 2 t 1 1 2 Constant Ricci Scalar 1 x 3 x 2 x Sphere S 3 + 1

The Line of n … Tough! Ricci scalar n=3 1 2 … n Tough! Ricci scalar n=3 1 2 1 2 3 Based on billions of Monte Carlo iterations! Complete dag so constant curvature!

X is the probability that the center node is on. Explode(n) Star (Naïve Bayes) n+1 1 2 3 n X is the probability that the center node is on. n > 1 Same vol. as Line of 3! Volumes increase very fast, n=2,3,…8 38.2522, 160.231, 698.646, 3121.57, 14178.0, 65157.6, 302107. Theorem: Exactly one arrow can be reversed without changing the total Volume!

Collapse(n) Star ? n d a b < R > 2 6 10 3 11 54 4 20 14 16 1/2 … n n+1 1 n d a b < R > 2 6 10 1/2 3 11 54 4 20 272 14 16 ?

Volume matters M compact, dimension d, volume V Minimum Description Length Generalization power of heat kernels

( Line(n+1), Explode(n) ) v/s Collapse(n) Explode > Line

B = symmetric beta function equal to a ratio of gammas A Fast Approx. Alg. The vol. of a dag of n nodes of dim. d is: Where: , p(j) is the prob that the parents of node i show jth pattern. Jensen’s inequality applied twice in two different ways shows: From where upper and lower bounds for the volume are obtained as: ai and bi are very simple sums that depend on the local topology of the dag. B = symmetric beta function equal to a ratio of gammas The approximations are exact for total disconnected and for complete dags. The sqrt(UL) is remarkably accurate as an estimate of Z… and it works!