HW2: Probability and Bayes Nets Due 10:00 AM, 04/29/2013 Please indicate the CEC ID of all group members clearly on the submission. Only one submission per group is required. Group size is not limited. 2.3 Conditional independence
Problem 1: Probability For the belief network shown below, indicate whether the following statements of conditional independence
Attach events to the binary random variables X, Y, and Z that are consistent with the are true (T) or false (F).
following patterns of common-­‐sense reasoning. You may use different events for the different parts of the problem. (An example of attaching an event to a binary random variable might be: “Roll two dice. Let X be the binary random variable that (a)
P (A|F ) = P (A)
is 1 if the sum of the dice is even and 0 otherwise, and Y the random variable that is 1 if at least also use sense variables like X=1 if (b) one die shows a “2”. You can P
(B|C,
I)common = P (B|I)
“it is raining” and Y=1 if “there is traffic”) (c)
P (F, G) = P (F ) P (G)
(a) Accumulating evidence: P(X=1) < P(X=1|Y 1) G)
< P(X=1|Y =1,Z=1) (d)
P (E,=F,
= P (E)
P (F ) P (G)
(b) Explaining away: (e)
P (C, D) = P (C) P (D)
P(X=1|Y =1) > P(X=1), P(X=1|Y =1,Z=1) < P(X=1|Y =1) P (D|I)
(f)
P (C, D|I)
= P (C|I)
(c) Conditional independence: (g)
P (C|A, B, D, E, F, H) = P (C)
P(X=1,Y =1) ≠ P(X=1)P(Y =1) (h)
P (A|B,
C, D, F,=G,
H, I) = P (A)
P(X=1,Y =1|Z=1) P(X=1|Z=1)P(Y =1|Z=1) (i)
P (D, E|I) = P (D|I) P (E|I)
Problem 2: Conditional Independence P (A, B, C, D|E, G)
(j)
=
P (A, B|E) P (C, D|G)
For the belief network shown below, indicate whether the following statements of conditional independence are true (T) or false (F). A
B
E
C
F
H
D
G
I
(a) P(A|F) = P(A) (b) P(B|C,I) = P(B|I) (c) P(F,G) = P(F) P(G) (d) P(E,F,G) = P(E) P(F) P(G) (e) P(C,D) = P(C) P(D) (f) P(C,D|I) = P(C|I) P(D|I) (g) P(C|A,B,D,E,F,H) = P(C) (h) P(A|B,C,D,F,G,H,I) = P(A) (i) P(D,E|I) = P(D|I) P(E|I) (j) P(A,B,C,D|E,G) = P(A,B|E) P(C,D|G) Problem 3: Markov Blankets For the graphical model shown in Problem 2, indicate the Markov blanket for the following variables. (a) A (b) B (c) D (d) E (e) G (f) I Problem 4: Variable Elimination and Likelihood Weighting Your friendly neighborhood TA, like most other TAs, is intent on world domination. His first step, obviously, was to build a graphical model, as shown in the figure below. The variables are: Graduate (G), Free Food (FF), The Force (TF), Knowledge (K), Money (M), Power (R) and World Domination (WD). All the variables are binary valued {T,F}. The conditional probability tables in the figure denote P(A = T | B=T …) and P(A = T | B=F …). (a) How likely is the TA to take over the world, if he manages to graduate? Compute P(WD = T|G = T) by variable elimination. (Report the ordering used and the factors produced after eliminating each variable for the query.) (b) We know the TA passes an important exam (K = T) and starts attending lots of lunch seminars around campus (FF = T) but is still a powerless TA (R = F). From the following samples, estimate the probability of imminent World Domination (WD = T) using Likelihood Weighting. TF, ¬G, ¬M, ¬WD ¬TF, ¬G, ¬M, ¬WD TF, G, ¬M, WD ¬TF, ¬G, M, WD ¬TF, G, M, ¬WD Problem 5: HMMs and Particle Filtering You are an interplanetary search and rescue expert who has just received an urgent message: a rover on Mercury has fallen and become trapped in Death Ravine, a deep, narrow gorge on the borders of enemy territory. You zoom over to Mercury to investigate the situation. Death Ravine is a narrow gorge 6 miles long, as shown below. There are volcanic vents at locations A and D, indicated by the triangular symbols at those locations. The rover was heavily damaged in the fall, and as a result, most of its sensors are broken. The only ones still functioning are its thermometers, which register only two levels: hot and cold. The rover sends back evidence E = hot when it is at a volcanic vent (A and D), and E = cold otherwise. There is no chance of a mistaken reading. The rover fell into the gorge at position A on day 1, so X1 = A. Let the rover's position on day t be Xt ∈ {A,B,C,D,E,F}. The rover is still executing its original programming, trying to move 1 mile east (i.e. right, towards F) every day. However, because of the damage, it only moves east with probability 0.5, and it stays in place with probability 0.5. Your job is to figure out where the rover is, so that you can dispatch your rescue-­‐bot. (c) Three days have passed since the rover fell into the ravine. The observations were (E1 = hot , E2 = cold , E3 = cold). What is P (X3 | hot1, cold2, cold3), the probability distribution over the rover's position on day 3, given the observations? You decide to attempt to rescue the rover on day 4. However, the transmission of E4 seems to have been corrupted, and so it is not observed. (d) What is the rover's position distribution for day 4 given the same evidence, P (X4 | hot1, cold2, cold3)? (e) All this computation is taxing your computers, so the next time this happens you decide to try approximate inference using particle filtering to track the rover. If your particles are initially in the top configuration shown below, what is the probability that they will be in the bottom configuration shown below after one day (after time elapses, but before evidence is observed)?