Probabilistic Reasoning Over
Time
Bayes Filter Implementations
Gaussian filters
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Guidelines and tips
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
Understanding Probabilistic State Estimation & Filtering: Bayes Filters & Reasoning, Study notes of Computer Science
An overview of probabilistic reasoning, specifically bayes filters, and their applications in state estimation and filtering. Topics include gaussian filters, markov assumption, causal vs. Diagnostic reasoning, combining evidence, and recursive bayesian updating. Examples are given using bayes filters for robot state estimation and measurement combination.
Typology: Study notes
Pre 2010
1 / 44
This page cannot be seen from the preview
Don't miss anything!
Related documents
Partial preview of the text
Download Understanding Probabilistic State Estimation & Filtering: Bayes Filters & Reasoning and more Study notes Computer Science in PDF only on Docsity!
Probabilistic Reasoning Over
Time
Bayes Filter Implementations
Gaussian filters
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 : 0
t t
t
t t
t^
x z p u z x z p
=
Application •
Filtering: Compute state given aseries of evidences
Prediction:
Smoothing or hindsight
Moste likely explanation
)
| (^
: 1 t
t^
e x p
)
|
(^
: 1 t
k t^
e
x p
t
k )
|
(^
: 1
<
t
k^
e
x p
)
|
(^
: 1
: 1
t
t^
e
x p
Another Example: State Estimation
Suppose a robot obtains measurement
z
What is
P(open|z)?
Example •^
P(z|open) = 0.
P(z|
open) = 0.
-^
P(open) = P(
open) = 0.
Example •^
P(z|open) = 0.
P(z|
open) = 0.
-^
P(open) = P(
open) = 0.
(^67). 0 2 3
(^5). 0 (^3). 0 (^5). 0 (^6). 0
(^5). 0 (^6). 0
) |
(
) ( ) | ( ) ( ) | (
) ( ) | ( ) | (
=
=
⋅
⋅
⋅
=
¬
¬
= z
open P
open p
open
z P
open p
open z P
open P
open z P
z
open P •^ z
raises the probability that the door is open.
Combining Evidence •^
Suppose our robot obtains anotherobservation
z
How can we integrate this newinformation?
More generally, how can we estimate^ P(x| z
...z 1
n^
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
K
K
K
K
Example: Second Measurement^ •^
P(z
|open) = 0.5 2
P(z
open) = 0.
-^
P(open|z
14
Example: Second Measurement^ •^
P(z
|open) = 0.5 2
P(z
open) = 0.
-^
P(open|z
(^625). 0
5 8
1 3 3 5 2 3 1 2
2 3 1 2
) | ( ) | ( ) | ( ) | (
) | ( ) | ( ) , | (
1
2
1
2
1
2
1 2
=
= ⋅
⋅
⋅
=
¬
¬
=^
z
open P
open
z P
z
open P
open z P
z
open P
open z P
z z
open P • z
2
lowers the probability that the door is open.
Bayes Filter Algorithm^ 1.
Algorithm
Bayes_filter
(^
Bel(x),d
):
2.
η=
0
If
d
is a perceptual data item
z
then
For all
x
do
For all
x
do
Else if
d
is an action data item
u
then
For all
x
do
Return
Bel’(x)
) ( ) | ( )
('
x Bel x z P
x
Bel
=
) (' x Bel
=
η η
) ('
) ('
1
x
Bel
x
Bel
−
η
' )' (
)' , | (
) ('
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
Bayes Filters are Familiar! •
Kalman filters
Particle filters
Hidden Markov models
Dynamic Bayesian networks
Partially Observable Markov DecisionProcesses (POMDPs)
1
1
1
(^
−
−
−
∫
t t t t t t t
t^
dx
x
Bel
x u x P x z P x
Bel
A Simple Example •
Assume you and your friend riding a boat
You are not good in navigation using stars• He is good in navigation using stars
You do a location measurement at time t(only one dimensional)
- Your estimate x(t) = z(me)• Variance sigma^2(t) = sigma^2(z)
A Simple Example (2) •
Your friend makes anothermeasuremet at the same time
- His estimate x(t) = z(my friend)=z(m_f)• Variance sigma^2(t) = sigma^2(z(m_f)