ps3_sol - 1 Introduction to Probabilistic Graphical Models...

This preview shows pages 1–2. Sign up to view the full content.

1 Introduction to Probabilistic Graphical Models December 13, 2007 Problem set 3 - Solutions Lecturer: Eran Segal 1. (a) Prove the following theorem: Let G 1 and G 2 be two graphs over X . If they have the same skeleton and the same set of v-structures then they are I-equivalent. Answer: Assume that G 1 and G 2 have the same skeleton and the same v-structures. First we assume that ( X Y | Z ) I ( G 1 ) and we show that ( X Y | Z ) I ( G 2 ) . By saying that the two graphs have the same skeleton we say that they have the same trails. Lets look on some trail between X X and Y Y in G 1 that given Z is inactive, and we will show that this trail in G 2 is inactive too. Consider two cases: i. The trail in G 1 is inactive because some (at least one) of the nodes on the trail that are not in a v–structure are observed (in Z ). Then clearly these nodes also blocks the trail in G 2 . ii. Otherwise, all nodes on the trail that are not in a v–structure are not observed (not in Z ), but then for some v-structure V i - 1 ,V i ,V i +1 on the trail, non of the descendents of V i (including V i itself) are observed. That is for every node V such that there is a directed path in G 1 from V i to V , all the nodes on the path are not observed. Consider such a directed path from such a

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 02/05/2010 for the course CS CS229 taught by Professor Eransegal during the Fall '07 term at École Normale Supérieure.

Page1 / 2

ps3_sol - 1 Introduction to Probabilistic Graphical Models...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online