Hw1 - Homework assignment 1 PDF

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Homework assignment 1...

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CSE 410/510pm: Probabilistic Methods in AI

Homework #1 Due in class on Wednesday, April 10, 2013 Guidelines: You can brainstorm with others, but please write up the answers by yourself. You may use textbooks (Koller & Friedman, Russell & Norvig, etc.), your notes, and lecture slides. Please show enough of your work to make your approach clear. 1. Given the following tables defining P (A|B, C), P (B), and P (C|B), compute P (A, B, C), P (A, C), and P (C|A). Express your answers as tables in the same format. A a0 a0 a0 a0 a1 a1 a1 a1

B b0 b0 b1 b1 b0 b0 b1 b1

C c0 c1 c0 c1 c0 c1 c0 c1

P (A|B, C ) 0.9 0.5 0.3 0.8 0.1 0.5 0.7 0.2

B b0 b1

P (B) 0.6 0.4

C c0 c0 c1 c1

B b0 b1 b0 b1

P (C|B ) 0.2 0.6 0.8 0.4

HINT: Use a calculator. When computing P (A, B, C), you should obtain P (a0 , b0 , c0 ) = 0.9 · 0.6 · 0.2 = 0.108 and P (a1 , b1 , c0 ) = 0.7 · 0.4 · 0.6 = 0.168 as two of the eight entries. If you do this step correctly, then the entries in P (A, B, C) should sum to one without requiring any additional normalization. If you get confused, ask for help! 2. Suppose the false positive rate of an antibody test used to diagnose HIV is 5/100,000, and that the false negative rate is negligible (you may assume it to be zero). In this context, “false positive rate” is the probability of a person who does not have HIV wrongly testing positive for HIV, and “false negative rate” is the probability of a person with HIV wrongly testing negative for HIV. Suppose that, out of 300 million people living in the US, 1 million have HIV. If a random person in the US is given this HIV test and receives a positive result, what is the probability that they actually have HIV? 3. K&F 2.3: Consider two events α and β such that P (α) = pa and P (β) = pb . Given only that knowledge, what is the maximum and minimum values of the probability of the events α ∩ β (“α AND β”) and α ∪ β (“α OR β”)? Characterize the situations in which each of these extreme values occurs. HINTS: Express your answers in terms of pa and pb . Give the tightest bounds you can. 1 is an obvious upper bound; can you do better? Draw Venn diagrams if it helps. 4. [Grads only] K&F 2.4: Let P be a distribution over (Ω, S), and let α ∈ S be an event such that P (α) > 0. The conditional probability P (·|α) assigns a value to each event in S. Show that it satisfies the properties of definition 2.1.

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