final-vpi-sols

CS 188 Spring 2023 Final Review: VPI Solutions Q1. Vehicle Perception Indication A vehicle is trying to identify the situation of the world around it using a set of sensors located around the vehicle. Each sensor reading (SRead) is based off of an object’s location (LOC) and an object’s movement (MOVE). The sensor reading will then produce various values for its predicted location (SLoc) and predicted movement (SMove). The user will receive these readings, as well as the the image (IMAGE) as feedback. (a) The vehicle takes an action, and we assign some utility to the action based on the object’s location and movement. Possible actions are MOVE TOWARDS, MOVE AWAY, and STOP. Suppose the decision network faced by the vehicle is the following. (i) Based on the diagram above, which of the following could possibly be true? ■ VPI (Image) = 0 □ VPI (SRead) < 0 □ VPI (SMove , SRead) > VPI (SRead) ■ VPI (Feedback) = 0 *$ None of the above VPI(Image) = 0 because there is not active path connecting Image and U VPI cannot be negative, so option 2 is not selected. VPI(SMove, SRead) = VPI(SMove | SRead) + VPI(SRead), therefore we can cancel VPI(SRead) from both side, and it becomes asking if VPI(SMove | SRead) > 0. And we can see that cannot be true, because shading in SRead, there is no active path connecting SMove and U. 1

There is an active path connecting Feedback and U, therefore VPI(Feedback) ≥ 0. It could still be 0 because active path only gives the possibility of > 0. (ii) Based on the diagram above, which of the following must necessarily be true? ■ VPI (Image) = 0 □ VPI (SRead) = 0 ■ VPI (SMove , SRead) = VPI (SRead) □ VPI (Feedback) = 0 *$ None of the above VPI(Image) = 0 because there is not active path connecting Image and U VPI(SRead) could be > 0 because SRead-MOVE-U is an active path between SRead and U VPI(SMove, SRead) = VPI(SMove | SRead) + VPI(SRead), therefore we can cancel VPI(SRead) from both side, and it becomes asking if VPI(SMove | SRead) == 0. And we can see that must true, because shading in SRead, there is no active path connecting SMove and U. VPI(Feedback) could be > 0 because feedback-SLoc-SRead-MOVE-U is an active path 2

Choice 2 is true because VPI(SLoc | SRead) = 0, and thus VPI(SRead) = VPI(SRead) + 0 = VPI(SRead) + VPI(SLoc | SRead) = VPI(SRead, SLoc), which makes choice 2 the same as choice 4 4

Q2. [Optional] Dressed to Impress (a) Alice is invited to a party tonight which is said to be once-in-a-lifetime. However, this mysterious party doesn’t publicize who is going and thus Alice has no idea whether the size S of the party will be be large ( S = + s ) or tiny ( S = − s ). The size can affect the noise N outside the party, and it will either be noisy ( N = + n ) or quiet ( N = − n ). Alice has three dresses: blue, red and yellow. Each dress will have a different utility for her depending on the size of the party. Let’s help her decide which will be best! We have the following decision network, where circles are chance nodes, squares are decision nodes, and diamonds are utility nodes: U A S N S P ( S ) +s 0.5 -s 0.5 S N P ( N | S ) +s +n 0.7 +s -n 0.3 -s +n 0.1 -s -n 0.9 S A U +s blue 80 -s blue 60 +s red 40 -s red 100 +s yellow 60 -s yellow 40 (i) What is the expected utility of wearing each dress, with both S and N unobserved? • EU(A=blue) = 80 ∗ 0 . 5 + 60 ∗ 0 . 5 = 70 • EU(A=red) = 40 ∗ 0 . 5 + 100 ∗ 0 . 5 = 70 • EU(A=yellow) = 60 ∗ 0 . 5 + 40 ∗ 0 . 5 = 50 What is Alice’s maximum expected utility? • MEU( {} ) = 70 (ii) Suppose Alice observes that the party is quiet, N = − n . Compute the following conditional proba- bilities with this observation: • P (+ s | − n ) = P ( − n | + s ) P (+ s ) P ( − n | + s ) P (+ s )+ P ( − n |− s ) P ( − s ) = 0 . 3 · 0 . 5 0 . 3 · 0 . 5+0 . 9 · 0 . 5 = 0 . 25 • P ( − s | − n ) = P ( − s | − n ) = 1 - P (+ s | − n ) = 0 . 75 What is the expected utility of wearing each dress? • EU(A=blue | N = − n ) = 80 ∗ 0 . 25 + 60 ∗ 0 . 75 = 145 2 = 65 • EU(A=red | N = − n ) = 40 ∗ 0 . 25 + 100 ∗ 0 . 75 = 155 2 = 85 • EU(A=yellow | N = − n ) = 60 ∗ 0 . 25 + 40 ∗ 0 . 75 = 45 What is Alice’s maximum expected utility given that N = − n ? • MEU( { N=-n } ) = 85 (iii) Construct a formula for V PI ( N ) for the given network. To decouple this problem from your work above, use any of the symbolic terms from the following list (rather than plugging in numeric values): P (+ n | + s ) , P (+ n | − s ) , P ( − n | + s ) , P ( − n | − s ) , P (+ n ) , P ( − n ) , P (+ s ) , P ( − s ), MEU ( {} ) , MEU ( { N = + n } ) , MEU ( { N = − n } ) • V PI ( N ) = P (+ n ) MEU ( { N = + n } ) + P ( − n ) MEU ( { N = − n } ) − MEU ( {} ) 5