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Introduction to Probabilistic Robotics: Uncertainty & Bayesian Reasoning

Learn about explicit representation of uncertainty using probability theory calculus, perception as state estimation, and action as utility optimization. Explore axioms, random variables, joint probabilities, and Bayesian reasoning. Understand the Bayes formula and normalization algorithm.

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Introduction to Probabilistic Robotics: Uncertainty & Bayesian Reasoning

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  1. Probabilistic Robotics Introduction Probabilities Bayes rule Bayes filters

  2. Probabilistic Robotics Key idea: Explicit representation of uncertainty using the calculus of probability theory • Perception = state estimation • Action = utility optimization

  3. Axioms of Probability Theory Pr(A) denotes probability that proposition A is true.

  4. B A Closer Look at Axiom 3

  5. Using the Axioms

  6. Discrete Random Variables • Xdenotes a random variable. • Xcan take on a countable number of values in {x1, x2, …, xn}. • P(X=xi), or P(xi), is the probability that the random variable X takes on value xi. • P( ) is called probability mass function. • E.g. .

  7. Continuous Random Variables • Xtakes on values in the continuum. • p(X=x), or p(x), is a probability density function. • E.g. p(x) x

  8. Joint and Conditional Probability • P(X=x and Y=y) = P(x,y) • If X and Y are independent then P(x,y) = P(x) P(y) • P(x | y) is the probability of x given y P(x | y) = P(x,y) / P(y) P(x,y) = P(x | y) P(y) • If X and Y are independent thenP(x | y) = P(x)

  9. Conditional Probability • P(A|B) = P(A and B) / P(B) 0.5 B P(A|B) = 0.2/0.5 A 0.2 Introduction to Robotics นัทที นิภานันท์ บทที่ 3 หน้า 9

  10. Law of Total Probability, Marginals Discrete case Continuous case

  11. Bayes Formula

  12. Bayesian reasoning • มีสมมุติฐาน h ที่ต้องการรู้ว่าเป็นจริงรึเปล่าเช่น “คนไข้เป็นมะเร็งรึเปล่า” • มีข้อมูล d ที่สังเกตได้ซึ่งเป็นผลจากสมมุติฐาน h • โดยทั่วไปเรามักจะรู้ P(d|h) • แต่เราอยากรู้ P(h|d) จะทำอย่างไร? (เราเรียกความน่าจะเป็นนี้ว่า posterior probability) Introduction to Robotics อรรถวิทย์ สุดแสง บทที่ 3 หน้า 12

  13. Bayesian reasoning ตัวอย่าง • สมมุติฐาน h แทนคนไข้เป็นมะเร็ง • ผลการตรวจ d มีสองกรณีคือ d+ (เป็น) และ d- (ไม่เป็น) • จากข้อมูลที่เก็บมาเรารู้ว่าความน่าจะเป็นที่คนหนี่งเป็นมะเร็งคือ P(h)=0.008 • จากการวิจัยพบว่าความแม่นยำของผลการตรวจเป็นดังนี้ • ถ้าเป็นมะเร็งจะตรวจได้ผลบวกด้วยความน่าจะเป็น P(d+| h)=0.98 • ถ้าไม่เป็นจะตรวจได้ผลลบด้วยความน่าจะเป็น P(d-| h)=0.97 Introduction to Robotics อรรถวิทย์ สุดแสง บทที่ 3 หน้า 13

  14. Bayesian reasoning ตัวอย่าง • ถ้ามีคนไข้มาตรวจและได้ผลเป็น d+ จะสรุปอย่างไรดี • เริ่มดูข้อมูลที่มี P(h) = 0.008 P(~h)=0.992 P(d+ | h)=0.98 P(d- | h)=0.02 P(d+ | ~h)=0.03 P(d- | ~h)=0.97 • ต้องการรู้ว่าจะสรุป P(h | d+) หรือ P(~h | d+) P(h | d+) = P(d+ | h) P(h) / P(d+) = 0.0078 / P(d+) P(~h | d+) = P(d+ | ~h) P(~h) / P(d+) = 0.0298 / P(d+) Introduction to Robotics อรรถวิทย์ สุดแสง บทที่ 3 หน้า 14

  15. Bayesian reasoning ตัวอย่าง • เพราะ P(h | d+) + P(~h | d+) = 1 จึงได้ว่า P(h | d+) = 0.21 และ P(~h | d+) = 0.79 สรุปว่าไม่เป็นมะเร็งน่าเชื่อถือมากกว่า • และก็ได้อีกว่า P(d+) = 0.0078/0.21 = 0.037 Introduction to Robotics อรรถวิทย์ สุดแสง บทที่ 3 หน้า 15

  16. Bayesian reasoning “judging” one’s intuition Scenario: Suppose a crime has been committed. Blood is found at the scene for which there is no innocent explanation. It is of a blood type which is present in 1% of the population and present in the defendant. There is a 1% chance that the defendant would have the crime scene’s blood type if innocent, so there is a 99% chance that the defendant is guilty. Prosecutor: Defendant: The crime occurred in a city of 800,000 people (suppose true). There, the blood type would be found in approximately 8,000 people, so this evidence only provides a negligible 0.0125% (= 1/8000) chance of guilt. Introduction to Robotics อรรถวิทย์ สุดแสง บทที่ 3 หน้า 16

  17. Normalization Algorithm:

  18. Conditioning • Law of total probability:

  19. Bayes Rule with Background Knowledge

  20. Conditioning • Total probability:

  21. Conditional Independence equivalent to and

  22. Simple Example of State Estimation • Suppose a robot obtains measurement z • What is P(open|z)?

  23. count frequencies! Causal vs. Diagnostic Reasoning • P(open|z) is diagnostic. • P(z|open) is causal. • Often causal knowledge is easier to obtain. • Bayes rule allows us to use causal knowledge:

  24. Example • P(z|open) = 0.6 P(z|open) = 0.3 • P(open) = P(open) = 0.5 • z raises the probability that the door is open.

  25. Combining Evidence • Suppose our robot obtains another observation z2. • How can we integrate this new information? • More generally, how can we estimateP(x| z1...zn )?

  26. Recursive Bayesian Updating Markov assumption: znis independent of z1,...,zn-1if we know x.

  27. Example: Second Measurement • P(z2|open) = 0.5 P(z2|open) = 0.6 • P(open|z1)=2/3 • z2lowers the probability that the door is open.

  28. A Typical Pitfall • Two possible locations x1and x2 • P(x1)=0.99 • P(z|x2)=0.09 P(z|x1)=0.07

  29. Actions • Often the world is dynamic since • actions carried out by the robot, • actions carried out by other agents, • or just the time passing by change the world. • How can we incorporate such actions?

  30. Typical Actions • The robot turns its wheels to move • The robot uses its manipulator to grasp an object • Plants grow over time… • Actions are never carried out with absolute certainty. • In contrast to measurements, actions generally increase the uncertainty.

  31. Modeling Actions • To incorporate the outcome of an action u into the current “belief”, we use the conditional pdf P(x|u,x’) • This term specifies the pdf that executing u changes the state from x’ to x.

  32. Example: Closing the door

  33. State Transitions P(x|u,x’) for u = “close door”: If the door is open, the action “close door” succeeds in 90% of all cases.

  34. Integrating the Outcome of Actions Continuous case: Discrete case:

  35. Example: The Resulting Belief

  36. Bayes Filters: Framework • Given: • Stream of observations z and action data u: • Sensor modelP(z|x). • Action modelP(x|u,x’). • Prior probability of the system state P(x). • Wanted: • Estimate of the state X of a dynamical system. • The posterior of the state is also called Belief:

  37. Markov Assumption Underlying Assumptions • Static world • Independent noise • Perfect model, no approximation errors

  38. Bayes Markov Total prob. Markov Markov z = observation u = action x = state Bayes Filters

  39. Bayes Filter Algorithm • Algorithm Bayes_filter( Bel(x),d ): • h=0 • Ifd is a perceptual data item z then • For all x do • For all x do • Else ifd is an action data item uthen • For all x do • ReturnBel’(x)

  40. Bayes Filters are Familiar! • Kalman filters • Particle filters • Hidden Markov models • Dynamic Bayesian networks • Partially Observable Markov Decision Processes (POMDPs)

  41. Summary • Bayes rule allows us to compute probabilities that are hard to assess otherwise. • Under the Markov assumption, recursive Bayesian updating can be used to efficiently combine evidence. • Bayes filters are a probabilistic tool for estimating the state of dynamic systems.

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