Introduction to Bayesian Estimation McGill COMP 765 Sept 12 th , - - PowerPoint PPT Presentation

Introduction to Bayesian Estimation

McGill COMP 765 Sept 12th, 2017

“Where am I?” – our first core problem

• Last class:
• We can model a robot’s motions and the world as spatial quantities
• These are not perfect and therefore it is up to algorithms to compensate
• Today:
• Representing motion and sensing probabilistically
• Formulation of localization as Bayesian inference
• Describe and analyze a first simple algorithm

Example: Landing on mars

Complementary input sources

• A GPS tells our global position, with constant

noise (+/- 5m): appears as jitter around path

• An IMU tell our relative motion, with

unknown yaw drift: diverges over time

• A good algorithm will fuse these inputs

intelligently and recover a path which best explains both:

• Smoother than GPS
• Less drift than IMU

Example: Self-driving

Source: Dave Ferguson “Solve for X” talk, July 2013 http://www.youtube.com/watch?v=KA_C6OpL_Ao

Raw Sensor Data

7

Measured distances for expected distance of 300 cm. It is crucial that we measure the noise well to integrate.

Sonar Laser

8

Probabilistic Robotics

Key idea: Explicit representation of uncertainty using the calculus of probability theory

• Perception = state estimation
• Action

= utility optimization

9

Discrete Random Variables

• X denotes a random variable.
• X can 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.

02 . , 08 . , 2 . , 7 . ) (  Room P

.

10

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

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

x p(x)

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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)

12

Law of Total Probability, Marginals





y

y x P x P ) , ( ) (





y

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





x

x P 1 ) (

Discrete case



1 ) ( dx x p

Continuous case



 dy y p y x p x p ) ( ) | ( ) (



 dy y x p x p ) , ( ) (

13

Bayes Formula evidence prior likelihood ) ( ) ( ) | ( ) ( ) ( ) | ( ) ( ) | ( ) , (       y P x P x y P y x P x P x y P y P y x P y x P

14

Normalization

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

1

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

x



   



 

y x x y x y x

y x P x x P x y P x

| | |

aux ) | ( : aux 1 ) ( ) | ( aux :       



Algorithm:

15

Conditioning

• Law of total probability:

  

   dz y z P z y x P y x P dz z P z x P x P dz z x P x P ) | ( ) , | ( ) ( ) ( ) | ( ) ( ) , ( ) (

16

Bayes Rule with Background Knowledge

) | ( ) | ( ) , | ( ) , | ( z y P z x P z x y P z y x P 

17

Conditioning

• Total probability:

  

   dz z P z y x P y x P dz z P z x P x P dz z x P x P ) ( ) , | ( ) ( ) ( ) | ( ) ( ) , ( ) (

18

Conditional Independence

) | ( ) | ( ) , ( z y P z x P z y x P  ) , | ( ) ( y z x P z x P  ) , | ( ) ( x z y P z y P 

equivalent to and

19

Simple Example of State Estimation

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

20

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:

) ( ) ( ) | ( ) | ( z P

• pen

P

• pen

z P z

• pen

P 

count frequencies!

21

Example

• P(z|open) = 0.6

P(z|open) = 0.3

• P(open) = P(open) = 0.5

67 . 3 2 5 . 3 . 5 . 6 . 5 . 6 . ) | ( ) ( ) | ( ) ( ) | ( ) ( ) | ( ) | (       

• 

 z

• pen

P

• pen

p

• pen

z P

• pen

p

• pen

z P

• pen

P

• pen

z P z

• pen

P

• z raises the probability that the door is open.

22

Combining Evidence

• Suppose our robot obtains another observation z2.
• How can we integrate this new information?
• More generally, how can we estimate

P(x| z1...zn )?

23

Recursive Bayesian Updating

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

1 1 1 1 1 1 1   



n n n n n n

z z z P z z x P z z x z P z z x P    

Assumption: zn is independent of z1,...,zn-1 if we know x.

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

... 1 ... 1 1 1 1 1 1 1 1

x P x z P z z x P x z P z z z P z z x P x z P z z x P

n i i n n n n n n n n



   

        

24

Example: Second Measurement

• P(z2|open) = 0.5

P(z2|open) = 0.6

• P(open|z1)=2/3

625 . 8 5 3 1 5 3 3 2 2 1 3 2 2 1 ) | ( ) | ( ) | ( ) | ( ) | ( ) | ( ) , | (

1 2 1 2 1 2 1 2

      

• 

 z

• pen

P

• pen

z P z

• pen

P

• pen

z P z

• pen

P

• pen

z P z z

• pen

P

• z2 lowers the probability that the door is open.

25

A Typical Pitfall

• Two possible locations x1 and x2
• P(x1)=0.99
• P(z|x2)=0.09 P(z|x1)=0.07
• Integrate same z repeatedly
• What are we doing wrong?

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 5 10 15 20 25 30 35 40 45 50 p( x | d) Number of integrations p(x2 | d) p(x1 | d)

26

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?

27

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.

28

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.

29

Example: Closing the door

30

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.

• pen

closed 0.1 1 0.9

31

Integrating the Outcome of Actions



 ' ) ' ( ) ' , | ( ) | ( dx x P x u x P u x P



 ) ' ( ) ' , | ( ) | ( x P x u x P u x P

Continuous case: Discrete case:

32

Example: The Resulting Belief

) | ( 1 16 1 8 3 1 8 5 10 1 ) ( ) , | ( ) ( ) , | ( ) ' ( ) ' , | ( ) | ( 16 15 8 3 1 1 8 5 10 9 ) ( ) , | ( ) ( ) , | ( ) ' ( ) ' , | ( ) | ( u closed P closed P closed u

• pen

P

• pen

P

• pen

u

• pen

P x P x u

• pen

P u

• pen

P closed P closed u closed P

• pen

P

• pen

u closed P x P x u closed P u closed P                  

 

33

Bayes Filters: Framework

• Given:
• Stream of observations z and action data u:
• Sensor model P(z|x).
• Action model P(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:

) , , , | ( ) (

1 1 t t t t

z u z u x P x Bel   } , , , {

1 1 t t t

z u z u d  

34

Markov Assumption

Underlying Assumptions

• Static world
• Independent noise
• Perfect model, no approximation errors

) , | ( ) , , | (

1 : 1 : 1 1 : 1 t t t t t t t

u x x p u z x x p

 



) | ( ) , , | (

: 1 : 1 : t t t t t t

x z p u z x z p 

1 1 1

) ( ) , | ( ) | (

  





t t t t t t t

dx x Bel x u x P x z P 

Bayes Filters

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

1 1 1 1 t t t t t

u z u x P u z u x z P    

Bayes z = observation u = action x = state

) , , , | ( ) (

1 1 t t t t

z u z u x P x Bel  

Markov

) , , , | ( ) | (

1 1 t t t t

u z u x P x z P   

Markov

1 1 1 1 1

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

  





t t t t t t t t

dx u z u x P x u x P x z P  

1 1 1 1 1 1 1

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

  





t t t t t t t t

dx u z u x P x u z u x P x z P   

Total prob. Markov

1 1 1 1 1 1

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

   





t t t t t t t t

dx z z u x P x u x P x z P  

36

Bayes Filter Algorithm

1.

Algorithm Bayes_filter( Bel(x),d ): 2. 0 3. If d is a perceptual data item z then 4. For all x do 5. 6. 7. For all x do 8. 9. Else if d is an action data item u then 10. For all x do 11. 12. Return Bel’(x)

) ( ) | ( ) ( ' x Bel x z P x Bel  ) ( ' x Bel    ) ( ' ) ( '

1

x Bel x Bel



 ' ) ' ( ) ' , | ( ) ( ' dx x Bel x u x P x Bel





1 1 1

) ( ) , | ( ) | ( ) (

  





t t t t t t t t

dx x Bel x u x P x z P x Bel 

We have the math, what’s left?

• Choose a data structure to represent Bel(x)
• Create update rules that integrate the motions and measurements
• Feed in data, and out come your results
• For the rest of this section, we will consider a huge variety of

methods that roughly fall within this framework. First examples:

• Fixed-resolution discretization of Bel : Markov Localization
• Variable-resolution discretization of Bel : Particle Filters
• Restriction of Bel to a simple parametric family : Kalman Filters

Discrete Bayes Filter Algorithm

1.

Algorithm Discrete_Bayes_filter( Bel(x),d ): 2. 0 3. If d is a perceptual data item z then 4. For all x do 5. 6. 7. For all x do 8. 9. Else if d is an action data item u then 10. For all x do 11. 12. Return Bel’(x)

) ( ) | ( ) ( ' x Bel x z P x Bel  ) ( ' x Bel    ) ( ' ) ( '

1

x Bel x Bel









'

) ' ( ) ' , | ( ) ( '

x

x Bel x u x P x Bel

Piecewise Constant Bel(x)

Piecewise Constant – What about Angle?

) , , (    y x x Bel

t

Implementation (1)

• To update the belief upon sensory input and to carry out

the normalization one has to iterate over all cells of the grid.

• Especially when the belief is peaked (which is generally the

case during position tracking), one wants to avoid updating irrelevant aspects of the state space.

• One approach is not to update entire sub-spaces of the

state space.

• This, however, requires to monitor whether the robot is de-

localized or not.

• To achieve this, one can consider the likelihood of the
• bservations given the active components of the state

space.

Implementation (2)

• To efficiently update the belief upon robot motions, one typically

assumes a bounded Gaussian model for the motion uncertainty.

• This reduces the update cost from O(n2) to O(n), where n is the

number of states.

• The update can also be realized by shifting the data in the grid

according to the measured motion.

• In a second step, the grid is then convolved using a separable

Gaussian Kernel.

• Two-dimensional example:

1/4 1/4 1/2 1/4 1/2 1/4

+ 

1/16 1/16 1/8 1/8 1/8 1/4 1/16 1/16 1/8

• Fewer arithmetic operations
• Easier to implement

43

Grid-based Localization

44

Sonars and Occupancy Grid Map

Conclusion:

• Bayesian estimation gives a framework to integrate evidence

assuming only local knowledge.

• The math tells us the correct operations to fuse the knowledge,

properly representing the posterior distribution

• Discrete Bayes Filters are never used: we must think of smarter ways

to represent our distributions and to compute updates (next time!)