ps3_sol - 1 Introduction to Probabilistic Graphical Models...

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

View Full Document Right Arrow Icon
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
Background image of page 1

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

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

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 Right Arrow Icon
Ask a homework question - tutors are online