Probabilistic Robotics: Understanding Uncertainty and Bayes Rule for Robotics, Slides of Robotics and Autonomous Systems

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Probabilistic Robotics

Introduction

Probabilities

Bayes rule

Bayes filters

Probabilistic Robotics

Key idea:

Explicit representation of uncertainty using the calculus of probability

theory

• Perception = state estimation
• Action = utility optimization

A Closer Look at Axiom 3

B

A A  B B

True

Pr( A  B )  Pr( A )Pr( B ) Pr( A  B )

Using the Axioms

Pr( ) 1 Pr( )

1 Pr( ) Pr( ) 0

Pr( ) Pr( ) Pr( ) Pr( )

Pr( ) Pr( ) Pr( ) Pr( )

A A

A A

True A A False

A A A A A A

  

   

   

       

Continuous Random Variables

• X takes on values in the continuum.
• p(X=x) , or p(x) , is a probability density function.
• E.g.

  

b

a

Pr( x ( a , b )) p ( x ) dx

x

p(x)

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 then

P(x | y) = P(x)

Bayes Formula

evidence

likelihood prior

P y

P y x P x

P x y

P x y P x y P y P y x P x

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



 

 

 



xy

x

xy

xy

x P x y

x P y x P x

|

|

|

: ( | ) aux

aux

1

:aux ( | ) ( )





 



 



Algorithm:

Bayes Rule

with Background Knowledge

P y z

P y x z P x z P x y z 

Conditioning

• Total probability:

P x y P x y z P z dz

P x P x z P z dz

P x P x z dz

Simple Example of State Estimation

• Suppose a robot obtains measurement z
• What is P(open|z)?

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:

P z

P z open P open P open z 

count frequencies!

Combining Evidence

• Suppose our robot obtains another observation z 2.
• How can we integrate this new information?
• More generally, how can we estimate

P(x| z 1 ...zn )?

Recursive Bayesian Updating

( | , , )

( | , , , ) ( | , , ) ( | , , ) 1 1

1 1 1 1 1 

   n n

n n n n P z z z

P z x z z P x z z P x z z 

  

Markov assumption : z n is independent of z 1 ,...,zn- 1 if

we know x.

( | ) ( )

( | ) ( | , , )

( | , , )

( | ) ( | , , ) ( | , , )

1 ...

1 ...

1 1

1 1

1 1 1

P z x P x

P z x P x z z

P z z z

P z x P x z z P x z z

i n

n i

n n

n n

n n n

















 



 