# homework2 - CS 440: Introduction to AI ...

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: CS 440: Introduction to AI  Homework 2  Due: Thursday September 24th    Your  answers  must  be  concise  and  clear.  Explain  sufficiently  that  we  can  easily  determine  what  you  understand.  We  will  give  more  points  for  a  brief  interesting  discussion  with  no  answer than for a bluffing answer.    Solutions  will  be  posted  no  sooner  than  two  days  after  the  due  date.  Homework  will  be  accepted until that point with a penalty of 10% per day that it is late.    No assignments will be  accepted after the solutions have been posted.   Late homework will only be accepted in class,  during office hours, or electronically by email to the TA.  You  are  expected  to  do  each  homework  on  your  own.  You  may  discuss  concepts  with  your  classmates, but there must be no interactions about solutions. You may consult the web but  the work handed in must be done on your own.  The  penalty  for  cheating  on  any  assignment  is  straightforward.  On  the  first  occurrence,  you  will receive a zero for the assignment and your course grade will be reduced by one full letter  grade. A second occurrence will result in course failure.    1) After your yearly checkup, the doctor has bad news and good news. The bad news is that you  tested  positive  for  a  serious  disease  and  that  the  test  is  99%  accurate  (i.e.,  the  probability  of  testing positive when you have the disease is 0.99 and the probability of testing negative when  you don’t have the disease is 0.99). The good news is that this is a rare disease, striking only 1 in  10,000 individuals. Why is it good news that the disease is rare? What are the chances that you  actually have the disease?    2)  Our research lab is having problems with his autonomous car and wants to model the system  using a Bayesian network. The objective is to avoid obstacles by reducing the speed of the car  and turning to drive around the obstacle. The variable ObstaclePresent denotes the presence of  an  obstacle  in  the  route.  If  an  obstacle  is  present  it  is  very  likely  that  the  obstacle  detection  system  will  detect  this  obstacle.  ObstacleDetected  determines  whether  and  obstacle  was  detected or not. Besides obstacles, the autonomous driver has to pay attention in cars coming  from  behind.  The  variable  CarBehind  indicates  has  value  true  if  there  is  a  car  behind  our  autonomous car. The variable CarDetected indicates that a car was detected behind our car, and  it is likely to be true if there is actually a car behind.   If  we  detect  an  obstacle  the  driver  is  likely  to  decide  to  turn  around,  as  well  as  it  is  likely  to  decide to reduce the speed. However, the driver is also likely to decide not to reduce the speed  if  it  detects  a  car  behind.  DecideToTurn  indicates  that  the  driver  decided  to  turn  and  DecideToReduce indicates that the driver decided to reduce the speed.   Currently, one of the big problems of our autonomous driver is the communication between the  decision components and the actuators. Even if the driver decides to reduce the speed there is  chance that the car will not reduce the speed and even if the driver decides to turn there is a  chance that the car will not turn. There is also a chance that the car will reduce speed or turn  without  a  decision  from  the  driver.  ReducedSpeed  indicates  that  the  car  actually  reduced  its  speed  while  Turned  indicates  whether  the  car  actually  turned  or  not.  It  is  also  important  to  remember that the car is likely to fail if it attempts to turn while driving in high speeds.   a) Use  the  ordering  ObstacleDetected,  ObstaclePresent,  Turned,  DecideToTurn,  CarBehind,  ReducedSpeed,  CarDetected  and  DecideToReduce  to  build  a  Bayesian  network to represent this problem. How many parameters are necessary represent the  conditional probabilities in this network?   b)  What  is  the  best  ordering  of  variables  that  you  can  find  for  this  problem?  Give  the  resulting  graphical  representation  of  the  Bayesian  network  and  say  how  many  parameters would we need at each node.                FellOﬀCliﬀ The robot fell oﬀ a cliﬀ. (a) Use the ordering ChangedDirection, DirtDetected, WallDetected, CliﬀDetected, DirtPresent, WallPresent, CliﬀPresent, DirtRemaining, BumpedIntoWall, FellOﬀCliﬀ to build a Bayesian network. (b) How many parameters are needed to represent the conditional probabilities in 1a? (c) The network in 1a may not be optimal. Give a Bayesian network so that the minimal number of parameters are needed to represent the conditional probabilities. (d) How many parameters are needed to represent the conditional probabilities in 1c?   3) Given the following Bayesian network of Boolean variables:  2. Given the following Bayesian network of Boolean variables: P (A) A (a) d d d d d © d  A P (B |A) A P (C |A) B C T (b) T (d) d F (c) F (e) d d d d c d © B P (D|B ) B C P (E |B, C ) D E T (f) TT (h) F (g) TF (i) FT (j) FF (k)   and the joint probability table   and the joint probabilities: B D ¬D E ¬E E ¬E C 0.00084 0.00336 0.00196 0.00784 ¬C 0.00378 0.03402 0.00882 0.07938 A C 0.00378 0.00162 0.00042 0.00018 ¬B B C 0.0096 0.0384 0.0224 0.0896 ¬C 0.03888 0.00972 0.00432 0.00108 1 ¬A ¬C 0.0048 0.0432 0.0112 0.1008 ¬B C ¬C 0.1512 0.1728 0.0648 0.0432 0.0168 0.0192 0.0072 0.0048 Compute the conditional probabilities (a)–(k). Show your work. 3. Given the following Bayesian network of Boolean variables:   a) Compute the probabilities a to k showing your work.   b)   A P (A) 0.4 A T F   B T F ii) Use the results from i) and a) to obtain a value for the probability.  © P (B |A) P (C ) B C 0.3 0.7 iii) Use the values from the joint to check that i) and ii) are correct.    d 0.5 d d d d c d © P (D|B ) B C P (E |B, C ) D E 0.8 TT 0.5 0.7 TF 0.2 FT 0.3 FF 0.6 i) Calculate P(E=T|A=T) (or P(e|a)) in terms of the probabilities  a to  k. (Use the letters  a  to k and not the number that they represent).  compute the following probabilities and show your work. Round your results to three digits after the decimal point if necessary. 4) Given the Bayesian network      a) Which  nodes  constitute  the  Markov  blanket  of  node  D?  To  which  nodes  D  is  conditionally independent given D’s Markov blanket?  b) Which  minimal  set  of  nodes  should  be  known  or  unknown  in  order  for  D  and  E  to  be  conditionally independent? Same thing for F and G.   c) Are the nodes B and C independent?  d) To which nodes E is conditionally independent given C? D or G?   ...
View Full Document

## This note was uploaded on 04/30/2011 for the course ECE 448 taught by Professor Levinson during the Spring '08 term at University of Illinois, Urbana Champaign.

Ask a homework question - tutors are online