10601 Machine Learning Assignment 7: Graphical Models
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1 060 Machine Learning Assignment 7: Graphical Models Due: Nov 3th, 4:29 EST, via Blackboard Late submission due: Nov 5th, 4:29 EST with 50% discount of credits TAs-in-charge: William Wang, Avi Dubey Policy on Collaboration among Students These policies are the same as were used in Dr Rosenfeld s previous version of 203 The purpose of student collaboration is to facilitate learning, not to circumvent it Studying the material in groups is strongly encouraged It is also allowed to seek help from other students in understanding the material needed to solve a particular homework problem, provided no written notes are shared, or are taken at that time, and provided learning is facilitated, not circumvented The actual solution must be done by each student alone, and the student should be ready to reproduce their solution upon request The presence or absence of any form of help or collaboration, whether given or received, must be explicitly stated and disclosed in full by all involved, on the first page of their assignment Specifically, each assignment solution must start by answering the following questions: id you receive any help whatsoever from anyone in solving this assignment? Yes / No If you answered yes, give full details: (eg Jane explained to me what is asked in Question 34 id you give any help whatsoever to anyone in solving this assignment? Yes / No If you answered yes, give full details: (eg I pointed Joe to section 23 to help him with Question 2 Collaboration without full disclosure will be handled severely, in compliance with CMU s Policy on Cheating and Plagiarism As a related point, some of the homework assignments used in this class may have been used in prior versions of this class, or in classes at other institutions Avoiding the use of heavily tested assignments will detract from the main purpose of these assignments, which is to reinforce the material and stimulate thinking Because some of these assignments may have been used before, solutions to them may be
2 (or may have been available online, or from other people It is explicitly forbidden to use any such sources, or to consult people who have solved these problems before You must solve the homework assignments completely on your own I will mostly rely on your wisdom and honor to follow this rule, but if a violation is detected it will be dealt with harshly Collaboration with other students who are currently taking the class is allowed, but only under the conditions stated below Which of the following statements on conditional independence are true? (X (Y, W Z implies (X Y Z True False 2 (X (Y, W Z implies ((X, W Y Z True False 3 (X (Y, W Z and (Y W Z implies ((X, W Y Z True False 2 X, X 2 and X 3 are random variables that take binary values {0, } Assume X and X 2 are drawn iid from Bernoulli( 2 (ie tossing a fair coin And X 3 = X XORX 2 In this case, which of the following statement is true? {X, X 2 } X 3 {X, X 2 } X 3 X X 2 2
3 None of the above 3 Assume we use a three-state HMM model to predict the next week s weather in Pittsburgh State : Snow State 2: Cloudy State 3: Sunny Assume that we have learned the following state transition probability matrix A in Fig and given that the weather on the day before next week is Sunny (State 3, what is the probability that the weather for next week being sun, sun, snow, snow, sun, cloudy, sun? Figure : The state transition probability matrix for HMM weather prediction model None of the above 4 Which of the statement on Hidden Markov Model (HMM and Conditional Random Fields (CRF is true? HMM is a generative model, while CRF is a discriminative model HMM models the conditional probability distribution of labels given the observations Pr(Y X, while CRF models the joint distribution of observations and labels Pr(X, Y 3
4 CRF does not have the Markov assumption All of the above 5 Bayesian Network Long time ago, your friendly 060 TA was intent on world domination, but now he just wants to graduate To identify his chance of graduating, he built a Bayesian Net model, shown in the Fig 2 The variables being: Graduate (G, which depends on papers (P and proposal (R As we know, papers and theorems (T are both generated by drinking lots of coffee (C And all grad students are dependent on free coffee provided by the university (U, which also allows our favorite coffee shop (D All the variables are binary valued {F, T } Figure 2: The Baysian Net for the TA graduation problem The CPT parameters are listed in the Fig 3: 5 What is the TA s chance to graduate if the coffee runs out? Compute P (G = T C = F =?
5 Figure 3: The conditional dependency table of the Bayesian Net 52 What is the TA s chance to graduate if Tazzo is open 24/7? Compute P (G = T D = T =? Is there a chance that the coffee runs out? Compute P (C = T =? What s the chance he will write papers, if we know he writes theorems? Compute P (P = T T = T =?
6 Figure 4: (a the graphical model of MRF (b the directions of BP with observed messages (c the known doubleton potentials 6 Markov Random Fields Belief Propagation is a popular inference algorithm in Markov Random Fields (MRFs To send a message, one multiplies all the incoming messages then multiple by the edge potential matrix and marginalize over the sender s note For example, if we want to send a message from node j to node i, we have: M j i (x i = Ψ ij (x i, x j Mj k (x j ( x j k Neighbors(j\i where as the Ψ ij (x i, x j is the doubleton potential on the edge between x i and x j To obtain the beliefs of a node, we can just multiple all the messages coming to that node: b j (x j = Mj k (x j (2 k Neighbors(j 6
7 Figure 4 shows the graphical model for the belief propagation problem Here, y and y 2 are observed variables, and we know the following initial messages and beliefs ( ( ( ( ( ( M y 4 =, M2 =, M3 2 =, M 2 =, M2 3 =, M y = (3 2 b = ( 4 6 ( 5, b 2 = 5 ( 8, b 3 = 2 (4 6 When we send a message from node to 2, what is the updated M 2? ( 9 ( 4 6 ( ( When we send a message from node 3 to 2, what is the updated M 3 2? ( 9 ( 8 2 ( 72 8 ( Now that we have updated M 2 and M 3 2, what is the updated b 2? Note: after computing the beliefs, you need to normalize the beliefs, so that they can sum to 7
8 ( 5 5 ( ( ( Now we keep sending messages from node 2 to 3, what is the updated M 2 3? ( ( ( ( 9 65 Now we send messages from node 2 to, what is the updated M 2? ( ( ( ( 9 8
9 66 Now that we have computed all the messages, what is the updated b? Note: after computing the beliefs, you need to normalize the beliefs, so that they can sum to ( 5 5 ( 4 6 ( 6 4 ( Now that we have computed all the messages, what is the updated b 3? Note: after computing the beliefs, you need to normalize the beliefs, so that they can sum to ( 5 5 ( ( 6 4 ( 8 2 9
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