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# Simple Benchmark in 3 dimensions

By Alice Cunningham,2014-05-05 21:37
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Simple Benchmark in 3 dimensions

Assignment 3+4

Dr. Eick’s COSC 6368 Fall 2001

Belief Networks and Logical Reasoning

Available Points: 45

Due Dates: Problems 6,7, and 8 are due on Nov. 20, 2001; problems 9 and 10 are due on Nov. 29, 2001.

6) Belief Networks, D-separability and Uncertainty [7]

Consider the following belief network that consists of variables A, B, C, D, E all of which have two states {true,false} and whose structure is depicted below is given.

B

A D E

C

a) Which of the following statements are implied by the indicated network structure;

i) P(A,D|C) = P(A|C)*P(D|C)

ii) P(B,C|A)=P(B|A)*P(C|A)

iii) P(E|A,D)=P(E|D)

b) Compare belief networks with “naïve Bayesian” classifiers and rule-based systems

that rely on making conditional independence assumption in their decision making.

What makes belief networks more “powerful”, if compared with “naïve Bayesian”

systems? Having used belief networks in the “astronomer”-problem what features of

belief network technology do you find attractive?

7) Belief Network Design [11]

Assume that John and Fred do not know each other, but take the same bus to come to UH to attend the same class; moreover, both use an alarm clock, and will be late to school if their alarm clock does not work; moreover, if the bus is late, both will come late to school. Assume you have to model this information using a belief network that consists of the following variables that have boolean ({true,false}) states:

; John-no-alarm: John’s alarm clock does not work (short JNOA)

; Fred-no-alarm: Fred’s alarm clock does not work (short FNOA)

; Bus-late: The bus that John and Fred take is late (short BL)

; John-late: John is late at school (short JL)

; Fred-late: Fred is late at school (short FL)

Moreover, assume that probability that John’s alarm clock does not work is 0.02, the probability that Fred’s alarm clock malfunctions is 0.02, and the probability that the bus is late is 0.1. You can also assume that Fred and John are on time if their alarm clock works and the bus they take isn’t late.

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a) Design the structure of a belief network that involves the above variables! Next

specify the probability tables for your network design (if there are probabilities

missing make up your own probabilities using common sense)

b) Using your results from the previous step, compute P(Bus-late=true|Fred-

late=true) by hand!!

8) Belief Networks (Paper & Pencil and/or Tool) -- 16 Points)

a) Solve subproblems a-c of the Telescope Problem (problem 15.3 of the textbook)! Give reasons for your answers when responding to the questions. Additionally, assume that the probabilty of a telescope being out of focus is 0.02, and that the probability of overcounting by one star is 0.05 and that the probability of undercounting by one star is 0.05; moreover, restrict you analysis to the case that N is limited to 0, 1, 2, 3, 4, and 5 stars! If there are any probabilities missing, just assume that they are evenly distributed (with respect to the set of states they might be in)!

b) Now assume we have 4 astronomers, in different parts of the world that make measurements M1, M2, M3, and M4 of the number of stars N in some region of the sky. Normally, there is a probability of 0.06 that the astronomer overlooks a star (undercounts by one star; you can assume that the four astronomers never overcount). Moreover, there is a 0.02 probability that a telescope is out of focus, in which the scientist undercounts by 3 or more stars (e.g. if N is 5 the scientist will count 2, 1 or 0 stars; you can assume that each case has the same probability). Design a belief network, and provide answers to the following problems (you are allowed to use Netica or another belief network tool to compute your answer):

; M1=3 M2=3 M3=1 M4=1

; M1=4 M2=1 M3=1 M4=1

; M1=5 M2=4 M3=3 M4=4

; N=5, M2=1

; M1=3 M2=3 M3=1

; M1=3 M2=2 M3=2

; N=4 F1=1 F2=1 F3=1

; N=3 F1=0 F2=0 F3=0 F4=0

9) First Order Predicate Logic [4]

Express the following natural language statements using first order predicate calculus formulas:

a) “There are at least two (different) students that take COSC 3480 [2]

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b) “Every person has a single social security number” (persons cannot have multiple

social security numbers)

c) “All persons that drive red cars are smart”

d) “Sally has at least on sister and no brother”

10) PROOFING THEOREMS USING RESOLUTION [7]

Prove using resolution (and not other methods!)

A1,A2,A3,A4,A5 |- A

with

(A1) "All red cars are owned by students”

(A2) "Everybody, who owns a red car, is rich."

(A3) "At least two students own a red car."

(A4) "Fred owns a blue car."

(A5) "Every student living in Houston owns at least one blue car" (A) "Some students are rich."

a) Transform the above natural language statements into first order predicate logic

formulas!

b) Convert the FOPL-formulas into clauses!

c) Show using resolution that the above statement is true

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