Winter 2008
Weekly Quiz
|
CS228: Probabilistic Methods in AI Winter 2008 Weekly Quiz |
| Overall Score Statistics: | ||||
|---|---|---|---|---|
| Mean | Median | Mode | Lowest | Highest |
| 85.25% | 90% | 100% (17) | 40% | 100% |
| Frequency of Scores | |
|---|---|
| Score | Number of Students |
| 100% | 17 |
| 90% | 16 |
| 80% | 13 |
| 70% | 10 |
| 60% | 1 |
| 50% | 1 |
| 40% | 1 |
Question 1: Which independencies hold in the following 2-TBN(Note, it may be helpful to draw the unfolded DBN for several slices):  Answers: 1. Weathert is independent of Velocityt given Observations1...t 2. Failuret is independent of Velocityt given Observations1...t 3. Weathert is independent of Velocityt given Weathert-1 and Observations1...t 4. Weathert is independent of Locationt given Velocityt and Observations1...t 5. None of the above | |||
|---|---|---|---|
| Type | Answer | Responses | Percent |
| Correct: | Weather<sup>t</sup> is independent of Velocity<sup>t</sup> given Weather<sup>t-1</sup> and Observations<sup>1...t</sup> | 48 | 81% |
| Distractor: | Weather<sup>t</sup> is independent of Velocity<sup>t</sup> given Observations<sup>1...t</sup> | 0 | 0% |
| Distractor: | Failure<sup>t</sup> is independent of Velocity<sup>t</sup> given Observations<sup>1...t</sup> | 1 | 2% |
| Distractor: | Weather<sup>t</sup> is independent of Location<sup>t</sup> given Velocity<sup>t</sup> and Observations<sup>1...t</sup> | 0 | 0% |
| Distractor: | None of the above | 10 | 17% |
Question 2: Which of the following template clique-trees allows for exact filtering in given 2-TBN?    Answers: 1. A 2. B 3. C 4. D 5. E 6. None of the above | |||
| Type | Answer | Responses | Percent |
| Correct: | A | 40 | 68% |
| Distractor: | B | 1 | 2% |
| Distractor: | C | 5 | 8% |
| Distractor: | D | 3 | 5% |
| Distractor: | E | 4 | 7% |
| Distractor: | None of the above | 6 | 10% |
| Question 3: What makes inference in DBNs difficult? Answers: 1. (a) As t grows large, we generally lose independence properties of the form (X(t) &perp Y(t) | Z(t)). 2. (b) Standard clique tree inference will not work in a DBN 3. (c) In many networks, maintaining a belief state over the variables requires representing a full joint 4. (b) and (c) 5. (a) and (c) | |||
| Type | Answer | Responses | Percent |
| Correct: | (a) and (c) | 50 | 85% |
| Distractor: | (a) As <i>t</i> grows large, we generally lose independence properties of the form <i>(X<sup>(t)</sup> &perp Y<sup>(t)</sup> &KPHHASH124; Z<sup>(t)</sup>)</i>. | 2 | 3% |
| Distractor: | (b) Standard clique tree inference will not work in a DBN | 0 | 0% |
| Distractor: | (c) In many networks, maintaining a belief state over the variables requires representing a full joint | 6 | 10% |
| Distractor: | (b) and (c) | 1 | 2% |
| Question 4: What is the key difference between particle filtering and likelihood weighting for DBNs? Answers: 1. In particle filtering, the particles are reweighed at each time step. 2. In particle filtering, the particles are resampled at each time step. 3. The particle filter performs exact inference at each time step. 4. Particle filtering starts with more particles. | |||
| Type | Answer | Responses | Percent |
| Correct: | In particle filtering, the particles are resampled at each time step. | 49 | 83% |
| Distractor: | In particle filtering, the particles are reweighed at each time step. | 9 | 15% |
| Distractor: | The particle filter performs exact inference at each time step. | 0 | 0% |
| Distractor: | Particle filtering starts with more particles. | 0 | 0% |
| Question 5: In general, if we initialize a bootstrap particle filter with a set of particles, where all particles are the same, i.e. x(0)[m] = x, for all m=1,...,M, then as the algorithm proceeds... Answers: 1. the belief state and all particles will remain unchanged, x(t)[m] = x. 2. the belief state will change, but all particles will be equal, x(t)[m] = x(t)[n]. 3. the belief state will change and the particles will become different, x(t)[m] != x(t)[n]. | |||
| Type | Answer | Responses | Percent |
| Correct: | the belief state will change and the particles will become different, <i>x<sup>(t)</sup>[m] != x<sup>(t)</sup>[n]</i>. | 50 | 85% |
| Distractor: | the belief state and all particles will remain unchanged, <i>x<sup>(t)</sup>[m] = x</i>. | 1 | 2% |
| Distractor: | the belief state will change, but all particles will be equal, <i>x<sup>(t)</sup>[m] = x<sup>(t)</sup>[n]</i>. | 8 | 14% |
| Question 6: Recall that the conditional entropy of X given Y is HP(X | Y) = -EP[log P(X | Y )]. Which of the following is true as we condition on more variables? Answers: 1. HP(X | Y, Z) <= HP(X | Y) 2. HP(X | Y, Z) >= HP(X | Y) 3. HP(X | Y, Z) = HP(X, Y) 4. Cannot be determined in general. | |||
| Type | Answer | Responses | Percent |
| Correct: | H<sub>P</sub>(X | Y, Z) <= H<sub>P</sub>(X | Y) | 54 | 92% |
| Distractor: | H<sub>P</sub>(X &KPHHASH124; Y, Z) >= H<sub>P</sub>(X &KPHHASH124; Y) | 3 | 5% |
| Distractor: | H<sub>P</sub>(X &KPHHASH124; Y, Z) = H<sub>P</sub>(X, Y) | 0 | 0% |
| Distractor: | Cannot be determined in general. | 2 | 3% |
| Question 7: D is the relative entropy. I(X ; Y) is equal to: Answers: 1. (a) D( P(X,Y) || P(X | Y )) 2. (b) D( P(X,Y) || P(X) P(Y) ) 3. (c) I(Y;X) 4. (a) and (c) 5. (b) and (c) | |||
| Type | Answer | Responses | Percent |
| Correct: | (b) and (c) | 41 | 69% |
| Distractor: | (a) D( P(X,Y) &KPHHASH124;&KPHHASH124; P(X &KPHHASH124; Y )) | 1 | 2% |
| Distractor: | (b) D( P(X,Y) &KPHHASH124;&KPHHASH124; P(X) P(Y) ) | 4 | 7% |
| Distractor: | (c) I(Y;X) | 7 | 12% |
| Distractor: | (a) and (c) | 6 | 10% |
Question 8: When does Bayesian prediction converge to the MLE estimate, using a Dirichlet prior? Recall that M is the total number of observed samples, and M' is the equivalent sample size, where  , for hyperparameters  .I. When M -> infinity II. When M' -> 0 Answers: 1. I 2. II 3. I and II 4. None of the above | |||
| Type | Answer | Responses | Percent |
| Correct: | I and II | 54 | 92% |
| Distractor: | I | 3 | 5% |
| Distractor: | II | 2 | 3% |
| Distractor: | None of the above | 0 | 0% |
| Question 9: Consider an experiment where we toss a thumbtack multiple times (independently). We model the probability distribution as
If we observe 20 heads and 30 tails, what is the MLE of theta? | |||
| Type | Answer | Responses | Percent |
| Correct: | 0.4 | 59 | 100% |
| Distractor: | 20 | 0 | 0% |
| Distractor: | 30 | 0 | 0% |
| Distractor: | 0.6 | 0 | 0% |
| Question 10: In the question above assume we now add a Dirichlet(25, 25) prior. What is the posterior estimate for theta? Answers: 1. 45 2. 55 3. 0.45 4. 0.55 5. | |||
| Type | Answer | Responses | Percent |
| Correct: | 0.45 | 58 | 98% |
| Distractor: | 45 | 1 | 2% |
| Distractor: | 55 | 0 | 0% |
| Distractor: | 0.55 | 0 | 0% |
| Distractor: | 0 | 0% | |
