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Probability: Review Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun , Burgard and Fox, Probabilistic R PowerPoint Presentation
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Probability: Review Pieter Abbeel UC Berkeley EECS Many slides adapted from Thrun , Burgard and Fox, Probabilistic Robotics. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A A A A A A A A. Why probability in robotics?.

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slide1
Probability: Review

Pieter Abbeel

UC Berkeley EECS

Many slides adapted from Thrun, Burgard and Fox, Probabilistic Robotics

TexPoint fonts used in EMF.

Read the TexPoint manual before you delete this box.: AAAAAAAAAAAAA

why probability in robotics
Why probability in robotics?
  • Often state of robot and state of its environment are unknown and only noisy sensors available
    • Probability provides a framework to fuse sensory information
    • Result: probability distribution over possible states of robot and environment
  • Dynamics is often stochastic, hence can’t optimize for a particular outcome, but only optimize to obtain a good distribution over outcomes
    • Probability provides a framework to reason in this setting
    • Result: ability to find good control policies for stochastic dynamics and environments
example 1 helicopter
Example 1: Helicopter
  • State: position, orientation, velocity, angular rate
  • Sensors:
    • GPS : noisy estimate of position (sometimes also velocity)
    • Inertial sensing unit: noisy measurements from
      • 3-axis gyro [=angular rate sensor],
      • 3-axis accelerometer [=measures acceleration + gravity; e.g., measures (0,0,0) in free-fall],
      • 3-axis magnetometer
  • Dynamics:
    • Noise from: wind, unmodeled dynamics in engine, servos, blades
example 2 mobile robot inside building
Example 2: Mobile robot inside building
  • State: position and heading
  • Sensors:
    • Odometry (=sensing motion of actuators): e.g., wheel encoders
    • Laser range finder:
      • Measures time of flight of a laser beam between departure and return
      • Return is typically happening when hitting a surface that reflects the beam back to where it came from
  • Dynamics:
    • Noise from: wheel slippage, unmodeled variation in floor
axioms of probability theory
Axioms of Probability Theory

Pr(A) denotes probability that the outcome ω is an element of the set of possible outcomes A. A is often called an event. Same for B.

Ω is the set of all possible outcomes.

ϕ is the empty set.

discrete random variables
Discrete Random Variables

x1

x2

  • 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., X models the outcome of a coin flip, x1 = head, x2 = tail, P( x1 ) = 0.5 , P( x2 ) = 0.5

x3

x4

.

continuous random variables
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

joint and conditional probability
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)
  • Same for probability densities, just P  p
law of total probability marginals
Law of Total Probability, Marginals

Discrete case

Continuous case

normalization
Normalization

Algorithm:

conditioning
Conditioning
  • Law of total probability:
conditional independence
Conditional Independence

equivalent to

and

simple example of state estimation
Simple Example of State Estimation
  • Suppose a robot obtains measurement z
  • What is P(open|z)?
causal vs diagnostic reasoning
Causal vs. Diagnostic Reasoning

count frequencies!

  • 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:
example
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.
combining evidence
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 )?
recursive bayesian updating
Recursive Bayesian Updating

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

example second measurement
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.
a typical pitfall
A Typical Pitfall
  • Two possible locations x1and x2
  • P(x1)=0.99
  • P(z|x2)=0.09 P(z|x1)=0.07