Probability - Advanced Robotics - Lecture Slides, Slides of Robotics

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Probability: Review

n Often state of robot and state of its environment are unknown and only noisy sensors available n Probability provides a framework to fuse sensory information à Result: probability distribution over possible states of robot and environment n Dynamics is often stochastic, hence can’t optimize for a particular outcome, but only optimize to obtain a good distribution over outcomes n Probability provides a framework to reason in this setting à Result: ability to find good control policies for stochastic dynamics and environments

Why probability in robotics?

n State: position and heading n Sensors: n Odometry (=sensing motion of actuators): e.g., wheel encoders n Laser range finder: n Measures time of flight of a laser beam between departure and return n Return is typically happening when hitting a surface that reflects the beam back to where it came from n Dynamics: n Noise from: wheel slippage, unmodeled variation in floor Example 2: Mobile robot inside building

5 n n n

Axioms of Probability Theory

0 ≤ Pr( A ) ≤ 1

Pr(!) = 1 Pr( A! B ) = Pr( A ) + Pr( B ) " Pr( A # B )

Pr( !) = 0

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.

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Using the Axioms

Pr( A !(" \ A )) = Pr( A ) + Pr(" \ A ) # Pr( A $(" \ A )) Pr(") = Pr( A ) + Pr(" \ A ) # Pr( !) 1 = Pr( A ) + Pr(" \ A ) # 0 Pr(" \ A ) = 1 # Pr( A )

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Discrete Random Variables

n X denotes a random variable. n X can take on a countable number of values in {x 1 , x 2 , …, x n }. n P(X=x i ) , or P(x i ) , is the probability that the random variable X takes on value x i . n P( ) is called probability mass function. n E.g., X models the outcome of a coin flip, x 1 = head, x 2 = tail, P( x 1 ) = 0.5 , P( x 2 ) = 0. . x 1 ! x 2 x 4 x 3

10 Joint and Conditional Probability n P(X=x and Y=y) = P(x,y) n If X and Y are independent then P(x,y) = P(x) P(y) n 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) n If X and Y are independent then P(x | y) = P(x) n Same for probability densities, just P à p

11 Law of Total Probability, Marginals

= y P ( x ) P ( x , y )

= y P ( x ) P ( x | y ) P ( y )

= x P ( x ) 1 Discrete case

p ( x ) dx = 1 Continuous case

p ( x ) = p ( x | y ) p ( y ) dy

p ( x ) = p ( x , y ) dy

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Normalization

( | ) ( ) 1 ( ) ( | ) ( ) ( ) ( | ) ( ) ( ) 1 P y x P x P y P y x P x P y P y x P x P x y x ∑ = = = = − η η x y x x y x y x P x y x P y x P x | | | : ( | ) aux aux 1 :aux ( | ) ( ) η η ∀ = = ∀ =

Algorithm:

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Conditioning

n Law of total probability: ∫ ∫ ∫ = = = P x y P x y z P z y dz P x P x z P z dz P x P x z dz ( ) ( | , ) ( | ) ( ) ( | ) ( ) ( ) ( , )

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Conditional Independence

P ( x , y z )= P ( x | z ) P ( y | z ) P ( x z )= P ( x | z , y ) P ( y z )= P ( y | z , x ) equivalent to and

17 Simple Example of State Estimation n Suppose a robot obtains measurement z n What is P(open|z)?

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Example

n P(z|open) = 0.6 P(z| ¬ open) = 0. n P(open) = P( ¬ open) = 0.

1. 67 3 2
2. 6 0. 5 0. 3 0. 5
3. 6 0. 5 ( | ) ( | ) ( ) ( | ) ( ) ( | ) ( ) ( | ) = = ⋅ + ⋅ ⋅ =

• ¬ ¬ = P open z P z open p open P z open p open P z open P open P open z

• z raises the probability that the door is open. P ( open | z ) = P ( z | open ) P ( open ) P ( z )

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Combining Evidence

n Suppose our robot obtains another observation z 2 . n How can we integrate this new information? n More generally, how can we estimate P(x| z 1 ...z n )?