CS 262A: Learning and Reasoning with Bayesian Networks Winter 2016 Assignment 2 - Due 11:55pm Wednesday, January 27 1. Consider the DAG in Figure 1: (a) List the Markovian assumptions asserted by the DAG. (b) Express Pr(a, b, c, d, e, f, g, h) in terms of network parameters. (c) Compute Pr(A = 0, B = 1) and Pr(E = 0 | A = 1). Justify your answers. (d) What is a MAP for variables A and B? Justify your answer. (e) True or false? Why? dsep(A, BH, E) dsep(G, D, A) dsep(G, F, A) dsep(AB, F, GH) (f ) Provide two Markov blankets for variable D. 2. Construct two distinct DAGs over variables A, B, C, and D. Each DAG must have exactly four edges, and the DAGs must agree on d-separation. 3. Identify a DAG that is a D-MAP for all distributions Pr over variables X. Similarly, identify another DAG that is an I-MAP for all distributions Pr over variables X. The last two questions should be done using SamIam.1 You can add screenshots to your solution file. 4. Joe's x-ray test comes back positive for lung cancer. The test's false negative rate is fn = .40 and its false positive rate is fp = .02. We also know that the prior probability of having lung cancer is c = .001. Describe a Bayesian network and a corresponding query for computing the probability that Joe has lung cancer given his positive x-ray. What is the value of this probability? Use sensitivity analysis to identify necessary and sufficient conditions on each of fn , fp , and c that guarantee the probability of cancer to be no less than 10% given a positive x-ray test. 5. We have three identical and independent temperature sensors that will trigger in: 90% of the cases where the temperature is high; 5% of the cases where the temperature is nominal; 1% of the cases where the temperature is low. The probability of high temperature is 20%, nominal temperature is 70%, and low temperature is 10%. Describe a Bayesian network and compute the following (after describing the corresponding queries): (a) Probability that the first sensor will trigger given that the other two sensors have also triggered. (b) Probability that the temperature is high given that all three sensors have triggered. (c) Probability that the temperature is high given that at least one sensor has triggered. 1 You can download SamIam from http://reasoning.cs.ucla.edu/samiam. 1 A B C D E F G A 1 0 H A .2 .8 B 1 0 A 1 1 1 1 0 0 0 0 B 1 1 0 0 1 1 0 0 B .7 .3 D 1 0 1 0 1 0 1 0 B 1 1 0 0 E 1 0 1 0 E|B .1 .9 .9 .1 D|AB .5 .5 .6 .4 .1 .9 .8 .2 Figure 1: A Bayesian network with some of its CPTs. Submission (1) You should submit your work on CCLE. (2) You should submit a formatted pdf file (no scans or pictures). (3) You can submit late work, subject to the following penalties: 25% off for up to one day; 50% off for up to two days; 100% off for more than 2 days. 2