Midterm II. Introduction to Artificial Intelligence. CS 188 Spring ˆ You have approximately 1 hour and 50 minutes.

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1 CS 188 Spring 2013 Introduction to Artificial Intelligence Midterm II ˆ You have approximately 1 hour and 50 minutes. ˆ The exam is closed book, closed notes except a one-page crib sheet. ˆ Please use non-programmable calculators only. ˆ Mark your answers ON THE EXAM ITSELF. If you are not sure of your answer you may wish to provide a brief explanation. All short answer sections can be successfully answered in a few sentences AT MOST. First name Last name SID EdX username First and last name of student to your left First and last name of student to your right For staff use only: Q1. Bayes Nets Representation /17 Q2. Bayes Net Reasoning /12 Q3. Variable Elimination /21 Q4. Bayes Net Sampling /14 Q5. Probability, Bayes Nets and Decision Networks /28 Q6. Perceptron /8 Total /100 1

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3 Q1. [17 pts] Bayes Nets Representation (a) [8 pts] Graph Structure: Conditional Independence Consider the Bayes net given below. A B C D E F G H I Remember that X Y reads as X is independent of Y given nothing, and X Y {Z, W } reads as X is independent of Y given Z and W. For each expression, fill in the corresponding circle to indicate whether it is True or False. (i) True False It is guaranteed that A B (ii) True False It is guaranteed that A C (iii) True False It is guaranteed that A D E (iv) True False It is guaranteed that A I E (v) True False It is guaranteed that B C I (vi) True False It is guaranteed that F A H (vii) True False It is guaranteed that D I {E, G} (viii) True False It is guaranteed that C H G 3

4 (b) Marginalization and Conditioning Consider a Bayes net over the random variables A, B, C, D, E with the structure shown below, with full joint distribution P (A, B, C, D, E). The following three questions describe different, unrelated situations (your answers to one question should not influence your answer to other questions). A B C D E (i) [3 pts] Consider the marginal distribution P (A, B, C, E) = d P (A, B, C, d, E), where D was eliminated. On the diagram below, draw the minimal number of arrows that results in a Bayes net structure that is able to represent this marginal distribution. If no arrows are needed write No arrows needed. A B C E (ii) [3 pts] Assume we are given an observation: A = a. On the diagram below, draw the minimal number of arrows that results in a Bayes net structure that is able to represent the conditional distribution P (B, C, D, E A = a). If no arrows are needed write No arrows needed. B C D E (iii) [3 pts] Assume we are given an observations: D = d. On the diagram below, draw the minimal number of arrows that results in a Bayes net structure that is able to represent the conditional distribution P (A, B, C, E D = d). If no arrows are needed write No arrows needed. A B C E 4

5 Q2. [12 pts] Bayes Net Reasoning P (A D, X) +d +x +a 0.9 +d +x a 0.1 +d x +a 0.8 +d x a 0.2 d +x +a 0.6 d +x a 0.4 d x +a 0.1 d x a 0.9 P (D) +d 0.1 d 0.9 P (X D) +d +x 0.7 +d x 0.3 d +x 0.8 d x 0.2 P (B D) +d +b 0.7 +d b 0.3 d +b 0.5 d b 0.5 (a) [3 pts] What is the probability of having disease D and getting a positive result on test A? P (+d, +a) = (b) [3 pts] What is the probability of not having disease D and getting a positive result on test A? P ( d, +a) = (c) [3 pts] What is the probability of having disease D given a positive result on test A? P (+d + a) = (d) [3 pts] What is the probability of having disease D given a positive result on test B? P (+d + b) = 5

6 Q3. [21 pts] Variable Elimination (a) [9 pts] For the Bayes net below, we are given the query P (A, E +c). All variables have binary domains. Assume we run variable elimination to compute the answer to this query, with the following variable elimination ordering: B, D, G, F. Complete the following description of the factors generated in this process: After inserting evidence, we have the following factors to start out with: P (A), P (B A), P (+c), P (D A, B, +c), P (E D), P (F D), P (G + c, F ) When eliminating B we generate a new factor f 1 as follows: f 1 (A, +c, D) = b P (b A)P (D A, b, +c) This leaves us with the factors: P (A), P (+c), P (E D), P (F D), P (G + c, F ), f 1 (A, +c, D) When eliminating D we generate a new factor f 2 as follows: This leaves us with the factors: When eliminating G we generate a new factor f 3 as follows: This leaves us with the factors: 6

7 When eliminating F we generate a new factor f 4 as follows: This leaves us with the factors: (b) [2 pts] Write a formula to compute P (A, E +c) from the remaining factors. (c) [2 pts] Among f 1, f 2, f 3, f 4, which is the largest factor generated, and how large is it? Assume all variables have binary domains and measure the size of each factor by the number of rows in the table that would represent the factor. (d) [8 pts] Find a variable elimination ordering for the same query, i.e., for P (A, E +c), for which the maximum size factor generated along the way is smallest. Hint: the maximum size factor generated in your solution should have only 2 variables, for a size of 2 2 = 4 table. Fill in the variable elimination ordering and the factors generated into the table below. Variable Eliminated Factor Generated For example, in the naive ordering we used earlier, the first row in this table would have had the following two entries: B, f 1 (A, +c, D). 7

8 Q4. [14 pts] Bayes Net Sampling Assume you are given the following Bayes net and the corresponding distributions over the variables in the Bayes net. P (C A, B) +c +a +b.25 -c +a +b.75 P (D C) P (A) P (B) +c -a +b.6 +d +c.5 A +a 0.1 +b.7 -c -a +b.4 -d +c.5 -a 0.9 -b.3 +c +a -b.5 +d -c.8 C D -c +a -b.5 -d -c.2 +c -a -b.2 -c -a -b.8 B (a) [2 pts] Assume we receive evidence that A = +a. If we were to draw samples using rejection sampling, on expectation what percentage of the samples will be rejected? (b) [6 pts] Next, assume we observed both A = +a and D = +d. What are the weights for the following samples under likelihood weighting sampling? Sample Weight (+a, b, +c, +d) (+a, b, c, +d) (+a, +b, c, +d) (c) [2 pts] Given the samples in the previous question, estimate P ( b + a, +d). (d) [4 pts] Assume we need to (approximately) answer two different inference queries for this graph: P (C + a) and P (C + d). You are required to answer one query using likelihood weighting and one query using Gibbs sampling. In each case you can only collect a relatively small amount of samples, so for maximal accuracy you need to make sure you cleverly assign algorithm to query based on how well the algorithm fits the query. Which query would you answer with each algorithm? Algorithm Query Algorithm Query Likelihood Weighting Gibbs Sampling Justify your answer: 8

9 Q5. [28 pts] Probability, Bayes Nets and Decision Networks It is Monday night, and Bob is finishing up preparing for the CS188 Midterm II that is coming up on Tuesday. Bob has already mastered all the topics except one: Decision Networks. He is contemplating whether to spend the remainder of his evening reviewing that topic (review), or just go to sleep (sleep). Decision Networks are either going to be on the test (+d) or not be on the test ( d). His utility of satisfaction is only affected by these two variables as shown below: (a) [5 pts] Maximum Expected Utility Compute the following quantities: EU(review) = D P(D) +d 0.5 -d 0.5 D A U(D,A) +d review d review 600 +d sleep 0 -d sleep 1500 EU(sleep) = MEU({}) = Action that achieves MEU({}) = 9

10 (b) [11 pts] The TA is on Facebook The TAs happiness (H) is affected by whether decision networks are going to be on the exam. The happiness (H) determines whether the TA posts on Facebook (+f) or doesn t post on Facebook ( f). The prior on D and utility tables remain unchanged. F H P (F H) +f +h 0.6 -f +h 0.4 +f -h 0.2 -f -h 0.8 H D P (H D) +h +d h +d h -d h -d 0.75 D P(D) +d 0.5 -d 0.5 D A U(D,A) +d review d review 600 +d sleep 0 -d sleep 1500 Decision network. Tables that define the model are shown above. H P (H) +h 0.6 -h 0.4 F P (F ) +f f 0.56 D F P (D F ) +d +f d +f d -f d -f F D P (F D) +f +d f +d f -d f -d D H P (D H) +d +h d +h d -h d -h 0.94 Tables computed from the first set of tables. Some of them might be convenient to answer the questions below. Compute the following quantities: EU(review + f) = EU(sleep + f) = MEU({+f}) = Optimal Action({+f}) = EU(review f) = EU(sleep f) = MEU({ f}) = Optimal Action({ f}) = V P I({F }) = 10

(c) VPI Comparisons Now consider the case where there are n TAs. Each TA follows the same probabilistic models for happiness (H) and posting on Facebook (F ) as in the previous question.

11 (c) VPI Comparisons Now consider the case where there are n TAs. Each TA follows the same probabilistic models for happiness (H) and posting on Facebook (F ) as in the previous question. (i) [3 pts] True False V P I(H 1 F 1 ) = 0 Justify: (ii) [3 pts] True False V P I(F 1 H 1 ) = 0 Justify: (iii) [3 pts] True False V P I(F 3 F 2, F 1 ) > V P I(F 2 F 1 ) Justify: (iv) [3 pts] True False V P I(F 1, F 2,..., F n ) < V P I(H 1, H 2,..., H n ) Justify: 11

12 Q6. [8 pts] Perceptron You have decided to become a teacher. The only issue is that you don t want to spend lots of time grading essays, so instead you decide to grade them all with a linear classifier. Your classifier considers the number of 7-letter (f 7 ) and 8-letter words (f 8 ) in an essay and then assigns a grade, either A or F, based on those two numbers. You have four graded essays to learn from: BIAS f 7 f 8 grade A (+) F (-) A (+) F (-) (a) [2 pts] You decide to run perceptron and being optimistic about the students essay writing capabilities, you decide to initialize your weight vector as (1, 0, 0). If the score from your classifier is greater than 0, it gives an A, if it is 0 or lower, it gives an F. Fill in the resulting weight vector after having seen the first training example and after having seen the second training example. BIAS f 7 f 8 Initial After first training example After second training example (b) [2 pts] True False The training data is linearly separable with the given features. Justify: (c) [4 pts] For each of the following decision rules, indicate whether there is a weight vector that represents the decision rule. If Yes then include such a weight vector. 1. A paper gets an A if and only if it satisfies (f 7 + f 8 7). Yes w = No 2. A paper gets an A if and only if it satisfies (f 7 5 AND f 8 4). Yes w = No 3. A paper gets an A if and only if it satisfies (f 7 5 OR f 8 4). Yes w = No 4. A paper gets an A if and only if it has between 4 and 6, inclusive, 7-letter words and between 3 and 5 8-letter words. Yes w = No 12

Midterm II. Introduction to Artificial Intelligence. CS 188 Spring ˆ You have approximately 1 hour and 50 minutes.

Midterm II. Introduction to Artificial Intelligence. CS 188 Spring ˆ You have approximately 1 hour and 50 minutes. CS 188 Spring 2013 Introduction to Artificial Intelligence Midterm II ˆ You have approximately 1 hour and 50 minutes. ˆ The exam is closed book, closed notes except a one-page crib sheet. ˆ Please use