Probability and Bayes' Theorem

  1. Consider the robot localization scenario described in these slides. What is the final probability distribution over room locations given the following series of sensor readings?

    b (done for you in the slides)
    a
    b

    Assume that the initial distribution is

    .4.4.1.1

    that the robot is not moving, and that sensor readings are independent given the robot's position. (I.e. one incorrect sensor reading does not increase the probability of seeing the same incorrect sensor reading in the future.)

    You should treat the distribution calculated after each sensor reading as the prior distribution for the next sensor reading.


  2. The following question is based on question 5 from Appendix A of Computational Principles of Mobile Robotics.
  3. A common technique for the localization of robots in industrial settings is to augment the local environment with specific landmarks (often visual) that can be sensed by the robot. Suppose we have such a landmark scheme. We are interested in two Boolean random variables: D - the robot has detected a landmark and P - there is actually a landmark present. Express each of the following in probability notation using these variables:
    • The system correctly identifies landmarks with probability .9.
      
      
      
      
    • Landmarks are identified falsely (i.e. when no landmark is present) with probability .01.
      
      
      
      
    • The probability of a landmark being present is .05.
      
      
      
      
    Answer the following questions by writing the desired quantity in probability notation, then solving for the solution.
    • What is the false alarm probability (probability of identifying a landmark when none is present)?
      
      
      
      
      
      
    • What is the missed detection probability (probability of missing a landmark even though one is present)?