4 proposed research projects
DESCRIPTION
4 Proposed Research Projects. SmartHome Encouraging patients with mild cognitive disabilities to use digital memory notebook for activities of daily living Diabetes Encouraging patients to use program and follow exercise advice to maintain glucose levels - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/1.jpg)
4 Proposed Research Projects• SmartHome
– Encouraging patients with mild cognitive disabilities to use digital memory notebook for activities of daily living
• Diabetes– Encouraging patients to use program and follow exercise advice to
maintain glucose levels• Machine Learning Curriculum Development
– Automatically construct a set of tasks such that it’s faster to learn the set than to directly learn the final (target) task
• Lifelong Machine Learning– Get a team of parrot drones and turtlebots to coordinate for a search
and rescue. Build a library of past tasks to learn the n+1st task faster
All at risk because of the government shutdown
![Page 2: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/2.jpg)
Thu
• Homework• (some?) labs• Contest : only 1 try– Will open classroom early
![Page 3: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/3.jpg)
• General approach:
• A: action• S: pose• O: observationPosition at time t depends on position previous position and action, and current observation
![Page 4: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/4.jpg)
Models of Belief
![Page 5: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/5.jpg)
Axioms of Probability Theory• denotes probability that proposition A is true.• denotes probability that proposition A is false.
1. 2. 3.
1)Pr(0 A
1)Pr( True 0)Pr( False
)Pr()Pr()Pr()Pr( BABABA
![Page 6: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/6.jpg)
A Closer Look at Axiom 3
B
BA A BTrue
)Pr()Pr()Pr()Pr( BABABA
![Page 7: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/7.jpg)
Using the Axioms to Prove
(Axiom 3 with )
(by logical equivalence)
∨
![Page 8: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/8.jpg)
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(xi) is called probability mass function.
• E.g. 2.0)( RoomP
![Page 9: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/9.jpg)
Continuous Random Variables• takes on values in the continuum.
• , or , is a probability density function.
• E.g.b
a
dxxpbax )()),(Pr(
x
p(x)
![Page 10: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/10.jpg)
Probability Density Function
• Since continuous probability functions are defined for an infinite number of points over a continuous interval, the probability at a single point is always 0.
x
p(x)Magnitude of curve could be greater than 1 in some areas. The total area under the curve must add up to 1.
![Page 11: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/11.jpg)
Joint Probability• Notation
• If X and Y are independent then
![Page 12: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/12.jpg)
Conditional Probability
• is the probability of x given y
• If X and Y are independent then
![Page 13: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/13.jpg)
Inference by Enumeration
=0.4
![Page 14: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/14.jpg)
Law of Total Probability
y
yxPxP ),()(
y
yPyxPxP )()|()(
x
xP 1)(
Discrete case
1)( dxxp
Continuous case
dyypyxpxp )()|()(
dyyxpxp ),()(
![Page 15: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/15.jpg)
Bayes Formula
)( yxp
evidenceprior likelihood
)()()|()(
yPxPxyPyxP
)()|()()|(),( xPxyPyPyxPyxP
Posterior (conditional) probability distribution
)( xyp
)(xp Prior probability distribution
If y is a new sensor reading:
Model of the characteristics of the sensor
)(yp
Does not depend on x
![Page 16: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/16.jpg)
Bayes Formula
evidenceprior likelihood
)()()|()(
yPxPxyPyxP
)()|()()|(),( xPxyPyPyxPyxP
x
xPxyPxPxyPyxP
)()|()()|()(
![Page 17: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/17.jpg)
Bayes Rule with Background Knowledge
)|()|(),|(),|(
zyPzxPzxyPzyxP
![Page 18: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/18.jpg)
Conditional Independence
equivalent to
and ),|()( xzyPzyP
),|()( yzxPzxP
)|()|(),( zyPzxPzyxP
![Page 19: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/19.jpg)
Simple Example of State Estimation• Suppose a robot obtains measurement • What is ?
![Page 20: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/20.jpg)
Causal vs. Diagnostic Reasoning
• is diagnostic.• is causal.• Often causal knowledge is easier to obtain.• Bayes rule allows us to use causal knowledge:
)()()|()|(
zPopenPopenzPzopenP
Comes from sensor model.
![Page 21: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/21.jpg)
Example P(z|open) = 0.6 P(z|open) = 0.3 P(open) = P(open) = 0.5
67.032
5.03.05.06.05.06.0)|(
)()|()()|()()|()|(
zopenP
openpopenzPopenpopenzPopenPopenzPzopenP
raises the probability that the door is open.
)()()|()|( zP
openPopenzPzopenP
![Page 22: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/22.jpg)
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 )?
![Page 23: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/23.jpg)
Recursive Bayesian Updating
),,|(),,|(),,,|(),,|(
11
11111
nn
nnnn
zzzPzzxPzzxzPzzxP
Markov assumption: zn is independent of z1,...,zn-1 if we know x.
)()()|()|( zP
openPopenzPzopenP
![Page 24: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/24.jpg)
Example: 2nd Measurement
• P(z2|open) = 0.5 P(z2|open) = 0.6• P(open|z1)=2/3
625.085
31
53
32
21
32
21
)|()|()|()|()|()|(),|(
1212
1212
zopenPopenzPzopenPopenzPzopenPopenzPzzopenP
lowers the probability that the door is open.
),,|(),,|()|(),,|(
11
111
nn
nnn
zzzPzzxPxzPzzxP
![Page 25: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/25.jpg)
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?
![Page 26: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/26.jpg)
Typical Actions• Actions are never carried out with absolute certainty.• In contrast to measurements, actions generally increase the
uncertainty. • (Can you think of an exception?)
![Page 27: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/27.jpg)
Modeling Actions
• To incorporate the outcome of an action u into the current “belief”, we use the conditional pdf
• This term specifies the pdf that executing changes the state from to .
![Page 28: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/28.jpg)
Example: Closing the door
![Page 29: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/29.jpg)
for :
If the door is open, the action “close door” succeeds in 90% of all cases.
open closed0.1 10.9
0
![Page 30: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/30.jpg)
Integrating the Outcome of Actions
')'()',|()|( dxxPxuxPuxP
)'()',|()|( xPxuxPuxP
Continuous case:
Discrete case:
Applied to status of door, given that we just (tried to) close it?
![Page 31: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/31.jpg)
Integrating the Outcome of Actions
')'()',|()|( dxxPxuxPuxP
)'()',|()|( xPxuxPuxP
Continuous case:
Discrete case:
P(closed | u) = P(closed | u, open) P(open) + P(closed|u, closed) P(closed)
![Page 32: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/32.jpg)
Example: The Resulting Belief
1615
83
11
85
109
)(),|()(),|(
)'()',|()|(
closedPcloseduclosedPopenPopenuclosedP
xPxuclosedPuclosedP
![Page 33: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/33.jpg)
Example: The Resulting Belief
1615
83
11
85
109
)(),|()(),|(
)'()',|()|(
closedPcloseduclosedPopenPopenuclosedP
xPxuclosedPuclosedP
OK… but then what’s the chance that it’s still open?
![Page 34: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/34.jpg)
Example: The Resulting Belief
)|(1161
83
10
85
101
)(),|()(),|(
)'()',|()|(
uclosedP
closedPcloseduopenPopenPopenuopenP
xPxuopenPuopenP
![Page 35: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/35.jpg)
Summary
• Bayes rule allows us to compute probabilities that are hard to assess otherwise.
• Under the Markov assumption, recursive Bayesian updating can be used to efficiently combine evidence.
![Page 36: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/36.jpg)
Quiz from Lectures Past
• If events a and b are independent,• p(a, b) = p(a) × p(b)
• If events a and b are not independent, – p(a, b) = p(a) × p(b|a) = p(b) × p (a|b)
• p(c|d) = p (c , d) / p(d) = p((d|c) p(c)) / p(d)
![Page 37: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/37.jpg)
Example
![Page 38: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/38.jpg)
Example
s
P(s)
![Page 39: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/39.jpg)
Example
![Page 40: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/40.jpg)
Example
How does the probability distribution change if the robot now senses a wall?
![Page 41: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/41.jpg)
Example 2
0.2 0.2 0.2 0.2 0.2
![Page 42: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/42.jpg)
Example 2
0.2 0.2 0.2 0.2 0.2
Robot senses yellow.
Probability should go up.
Probability should go down.
![Page 43: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/43.jpg)
Example 2
• States that match observation• Multiply prior by 0.6
• States that don’t match observation• Multiply prior by 0.2
0.2 0.2 0.2 0.2 0.2
Robot senses yellow.
0.04 0.12 0.12 0.04 0.04
![Page 44: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/44.jpg)
Example 2
!
The probability distribution no longer sums to 1!
0.04 0.12 0.12 0.04 0.04
Normalize (divide by total)
.111 .333 .333 .111 .111
Sums to 0.36
![Page 45: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/45.jpg)
Nondeterministic Robot Motion
R
The robot can now move Left and Right.
.111 .333 .333 .111 .111
![Page 46: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/46.jpg)
Nondeterministic Robot Motion
R
The robot can now move Left and Right.
.111 .333 .333 .111 .111
When executing “move x steps to right” or left:.8: move x steps.1: move x-1 steps.1: move x+1 steps
![Page 47: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/47.jpg)
Nondeterministic Robot Motion
Right 2
The robot can now move Left and Right.
0 1 0 0 0
0 0 0.1 0.8 0.1
When executing “move x steps to right”:.8: move x steps.1: move x-1 steps.1: move x+1 steps
![Page 48: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/48.jpg)
Nondeterministic Robot Motion
Right 2
The robot can now move Left and Right.
0 0.5 0 0.5 0
0.4 0.05 0.05 0.4 (0.05+0.05)0.1
When executing “move x steps to right”:.8: move x steps.1: move x-1 steps.1: move x+1 steps
![Page 49: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/49.jpg)
Nondeterministic Robot Motion
What is the probability distribution after 1000 moves?
0 0.5 0 0.5 0
![Page 50: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/50.jpg)
Example
![Page 51: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/51.jpg)
ExampleRight
![Page 52: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/52.jpg)
Example
![Page 53: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/53.jpg)
ExampleRight
![Page 54: 4 Proposed Research Projects](https://reader035.vdocuments.mx/reader035/viewer/2022070419/56815b60550346895dc947ce/html5/thumbnails/54.jpg)
Localization
Sense Move
Initial Belief
Gain Information Lose
Information