10708 Graphical Models: Homework 4 Solutions
1
Markov Network Representations
1. We can convert the triangle graph on (A, B, C) with potential (A, B, C) into a pairwise Markov Random Field by introducing a variable X and connecting the nodes as sho

10-725 Optimization, Spring 2008: Homework 4 Solutions
Due: Wednesday, April 2, beginning of the class
1
Lagrangian relaxation of Boolean LP [Han, 12 points]
1. [6 pts] The Lagrangian is L(x, , v) = cT x + T (Ax - b) - T x + xT diag()x = x diag()x

Probablistic Graphical Models, Spring 2007 Homework 2
Due at the beginning of class on 10/17/07
Instructions
There are six questions in this homework. The last question involves some programming which should be done in MATLAB. Do not attach your cod

10-725 Optimization, Spring 2008: Homework 1 Solutions
Due: Wednesday, February 6, beginning of the class
1
L -Regularized Regression [Han, 5 points]
1
2
Art Class [Gaurav, 10 points]
This question is designed to help you visualize the geometry

10708 Graphical Models: Homework 1
Due October 3rd, beginning of class
September 20, 2005
1
1.1
[15 pts] Conditional Probability
[5 pts]
Let X , Y, Z be three disjoint sets of variables such that S = X Y Z. Prove that P |= (X Y|Z) if and only

10-725 Optimization, Spring 2008: Homework 5 Solutions
Due: Wednesday, April 23, beginning of the class
(Acknowledgement: The solutions for Q1 and Q2 are adapted from Prasanna Veglagapudi)
1
Conjugate functions [Han, 20 points]
Derive the conjugat

Probablistic Graphical Models, Spring 2007 Homework 4 solutions
1
Importance Sampling
1. Why is computing the probability of a complete instantiation of the variables in an MRF computationally intractible? The probability of a complete instantiatio

Probablistic Graphical Models, Spring 2007 Homework 3 solutions
1
Score Equivalence
Solution due to Steve Gardiner Proposition 1. The Bayesian score with a K2 prior is not score equivalent Proof. Consider the problem of learning a Bayesian network

Probablistic Graphical Models, Spring 2007 Homework 1 solutions
1
Representation
Solution due to Steve Gardiner Consider N +1 binary random variables X1 , . . . XN , Y that have the following conditional independencies: i, jXi Xj | Y 1. Consider

10-708 Homework 5 Solutions
Problem 1: Gaussian Graphical Models [25 pts]
Gaussian graphical model over three variables x1 , x2 , x3 with 1 1 P1 , x2 , x3 T x exp 1 x x 3 2 1 2 2 2
(1)
where x 1 , x2 , x3 x and 1 , , column

10-708 Probabilistic Graphical Models Homework 1 Solutions
Adapted from solutions of Anton Chechetka
1
1.1
It is known that P |= ( |) P ( |) = P (|)P (|) Now, if P (, , ) = f (, )g(, ) then P (|) =
(f (, )g(, ) , (f (, )g(, )
(1) (2)
, P (|) =

10708 Graphical Models: Homework 1 Solutions
October 13, 2006
1
1.1
Conditional Probability
Prove P (S) = f (X, Z)g(Y, Z) = (X Y |Z) P (X, Y |Z) = P (X, Y, Z) = P (Z) P (X, Y, Z) = x y P (x, y, Z) f (X, Z)g(Y, Z) = P (X|Z)P (Y |Z) x f (x, Z) y g

10708 Graphical Models: Homework 4
Due November 15th, beginning of class
October 27, 2006
Instructions: There are six questions on this assignment. Each question has the name of one of the TAs beside it, to whom you should direct any inquiries regar

10708 Graphical Models: Homework 2 Solutions
1 I-equivalence
1.1
We want to show that two graphs G1 and G2 are I-equivalent if 1) they have the same trails, and 2) a trail is active in G1 iff it is active in G2 . It is easy to see that two graphs ha

10708 Graphical Models: Homework 2
Due October 11th, beginning of class
September 27, 2006
Instructions: There are seven questions on this assignment. Each question has the name of one of the TAs beside it, to whom you should direct any inquiries re

10708 Graphical Models: Homework 3 Solutions
1 Triangulation
1. The moralized Bayes net (figure 1) is produced by marrying the parents and dropping direction on DAG edges.
A B
C
D
E
F
G
H
Figure 1: Moralized Bayes Net 2. A, B, G, H, D, E, F,

10708 Graphical Models: Homework 3
Due October 27th, beginning of class
October 13, 2006
Instructions: There are six questions on this assignment. Each question has the name of one of the TAs beside it, to whom you should direct any inquiries regard

10-708 Probabilistic Graphical Models Homework 2 Solutions
Adapted from solutions of Anton Chechetka
1
By the decomposition property, we can assume X Y = V . Also, from the Running Intersection Property (RIP), we have that X Y Sij This is because

10-725 Optimization, Spring 2008: Homework 3 Solutions
April 2, 2008
1
Convexity [Han, 30 points]
(Acknowledgement: The solutions for question 1 are adapted from the submitted solution of Ajit Paul Singh)
1.1
Linear Maps between convex sets
Ass

10-725 Optimization, Spring 2008: Homework 2 Solutions
Due: Wednesday, February 25, beginning of the class
1
Vertex Cover [Gaurav, 25 points]
The goal of this problem is to illustrate the use of LPs for approximating NP-hard optimization problems.

Probablistic Graphical Models, Spring 2007 Homework 1
Due at the beginning of class on 10/08/07
Instructions
There are six questions in this homework. The last question involves programming which should be done in MATLAB. Do not attach your code to

10708 Graphical Models: Homework 2
Due October 17th, beginning of class
October 3, 2005
1
[15 pts] Clique Tree I-maps
In order to formalize the relationship between clique trees and Bayesian Networks, in this question you will prove that if P fac

Probablistic Graphical Models, Spring 2007 Homework 4
Due at the beginning of class on 11/26/07
Instructions
There are four questions in this homework. The last question involves some programming which should be done in MATLAB. Do not attach your co

Probablistic Graphical Models, Spring 2007 Homework 3
Due at the beginning of class on 11/05/07
Instructions
There are five questions in this homework. The last question involves some programming which should be done in MATLAB. Do not attach your co

Probablistic Graphical Models, Spring 2007 Homework 2 solutions
December 6, 2007
1
Markov Networks
Strong Union: XY |Z Transitivity: (XA|Z) (AY |Z) = (XY |Z) (2) = XY |Z, W (1)
Solution due to Yongjin Park
1.
See Figure 1.
Figure 1: BN G of

10708 Graphical Models: Homework 5
Due November 29th, beginning of class
November 16, 2006
Instructions: There are four questions on this assignment. Each question has the name of one of the TAs beside it, to whom you should direct any inquiries reg