cs221-ps4

cs221-ps4 - CS221 Problem Set #4 1 CS 221 Problem Set #4...

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CS221 Problem Set #4 1 CS 221 Problem Set #4 — Bayes Nets and Perception Due by 9:30am on Tuesday, December 1. Please see the course information page on the class website for late homework submission instructions. SCPD students can also fax their solutions to (650) 725-1449 or e-mail to cs221qa@cs.stanford.edu. 1 Written part (70 points) NOTE: These questions require thought, but do not require long answers. Please try to be as concise as possible. 1. [22 points] I-Equivalence Given two Bayesian networks over the same variables, bn 1 and bn 2 , bn 1 and bn 2 are said to be I-equivalent if all d-separation properties of bn 1 also hold for bn 2 and vice versa. (a) [3 points] In the Fgure below, which of the four networks are I-equivalent? C B (d) (c) (a) (b) C C B C B B A A A A (b) [5 points] In the network pictured below, enumerate all of the conditional inde- pendencies represented by the network. Each independence should be of the form I ( X,Y | Z ) where X and Y are variables and Z is a (possibly empty) set of variables. Note that if your list contains I ( X,Y | Z ) then it need not contain I ( Y,X | Z ). (Hint: There are 5: 1 non-conditional, 2 conditioned on a single variable, and 2 conditioned on 2 variables.) A C D B
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CS221 Problem Set #4 2 (c) [8 points] Is there another I-equivalent network to this network (except itself)? If so, draw the equivalent network. If not, briefy explain using high level reasoning based on the independencies why no other network over these four variables can be equivalent. (Hint: certain independencies that you listed in (b) will allow you to eliminate, in one shot, many of the possible structures as clearly being not I-equivalent to this one. Thus, your answer could have the following format: list an independence assertion from part (b), and conclude certain properties that the structure will have to satisfy in order to make this independence assertion true. Repeat this process until you have found a structure satisfying all the independencies, or until you show that none can exist.) (d) [6 points] Now consider the network pictured below. B C D A There are other possible networks that are I-equivalent to the network in 1d. Draw one such equivalent network. ( Hint: it might help to Frst enumerate all of the conditional independencies represented by the network.) 2. [12 points] InFerence in Bayesian Networks Consider the following Bayesian network:
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cs221-ps4 - CS221 Problem Set #4 1 CS 221 Problem Set #4...

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