Belief networks Conditional independence Syntax and semantics Exact inference Approximate inference...
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Transcript of Belief networks Conditional independence Syntax and semantics Exact inference Approximate inference...
CS 460, Belief Networks 1
Belief networks
Conditional independenceSyntax and semanticsExact inferenceApproximate inference
Mundhenk and Itti 2008. Based on material from S Russel and P Norvig
CS 460, Belief Networks 2
Independence
CS 460, Belief Networks 3
Conditional independence
CS 460, Belief Networks 4
Conditional Independence
Cavity
Toothache Catch
Other interactions may exist, but they are either insignificant, unknown or irrelevant. We leave them out.
CS 460, Belief Networks 5
Conditional Independence
Cavity
Toothache Catch
We assume that a “catch” is not influenced by a toothache and visa versa.
CS 460, Belief Networks 6
Conditional independence
CS 460, Belief Networks 7
Conditional Independence
Cavity
Toothache
Catch
(1a) Since Catch does not affect Toothache P(Toothache|Catch,Cavity) = P(Toothache|Cavity)
Cavity
Toothache
=
CS 460, Belief Networks 8
Conditional Independence
Cavity
Toothache Catch
(1b) Algebraically these statements are equivalent P(Toothache,Catch|Cavity) = P(Toothache|Cavity)P(Catch|Cavity)
Cavity
Toothache Catch
Cavity
=
CS 460, Belief Networks 9
Conditional independence
CS 460, Belief Networks 10
Belief networks
CS 460, Belief Networks 11
Example
Probabilities derivedfrom prior observations
CS 460, Belief Networks 12
Typical Bayesian Network
Burglary Earthquake
Alarm
JohnCalls MaryCalls
P(B)
0.001
P(E)
0.002
B E P(A)
T T 0.95
T F 0.94
F T 0.29
F F 0.001
A P(J)
T 0.90
F 0.05
A P(M)
T 0.70
F 0.01
Here we see both the Topology and the Conditional Probability Tables (CPT).
CS 460, Belief Networks 13
Semantics
CS 460, Belief Networks 14
Semantics
CS 460, Belief Networks 15
Markov blanket
CS 460, Belief Networks 16
Constructing belief networks
CS 460, Belief Networks 17
Burglary Earthquake
Alarm
JohnCalls MaryCalls
P(B)
0.001
P(E)
0.002
B E P(A)
T T 0.95
T F 0.94
F T 0.29
F F 0.001
A P(J)
T 0.90
F 0.05
A P(M)
T 0.70
F 0.01
What is the probability that the alarm has sounded but neither a burglary nor earthquake has occurred and both John and Mary call? – P j m a b e
Example: Full Joint Distribution
CS 460, Belief Networks 18
Burglary Earthquake
Alarm
JohnCalls MaryCalls
P(B)
0.001
P(E)
0.002
B E P(A)
T T 0.95
T F 0.94
F T 0.29
F F 0.001
A P(J)
T 0.90
F 0.05
A P(M)
T 0.70
F 0.01
0.9 0 0.990.7
0.00
.0 0.9980
2
1 9
06
0 0
PPP m aP j bP j a P eb em aa b e
CS 460, Belief Networks 19What if we find a new variable?
Burglary Earthquake
Alarm
JohnCalls MaryCalls
P(B)
0.001
P(E)
0.002
B S E P(A)
T T T 0.98
T T F 0.94
T F T 0.96
T F F 0.95
F T T 0.45
F T F 0.25
F F T 0.29
F F F 0.001
A P(J)
T 0.90
F 0.05
A P(M)
T 0.70
F 0.01
We find that storms can also set off alarms. We add that into our CPT. Notice that JohnCalls and MaryCalls stay the same since Storms were always there but were just unaccounted for. John and Mary did not change! However, we have better precision at P(A).
StormP(S)
0.1
CS 460, Belief Networks 20
What if we cause a new variable? (or a new variable just occurs)
Burglary Earthquake
Alarm
JohnCalls MaryCalls
P(B)
0.001
P(E)
0.002
B C E P(A)
T T T 0.98
T T F 0.94
T F T 0.96
T F F 0.95
F T T 0.45
F T F 0.25
F F T 0.29
F F F 0.001
A P(J)
T 0.90
F 0.05
A P(M)
T 0.70
F 0.01
What if we inject a new cause that was not there before. We pay a crazy guy to set off the alarm frequently, JohnCalls and MaryCalls may no longer be valid since we may have changed the behaviors. For instance the alarm goes off so often now that John and Mary are more likely to ignore it.
CrazyGuyP(C)
0.25
CS 460, Belief Networks 21
What if we cause a new variable?
Burglary Earthquake
Alarm
JohnCalls MaryCalls
P(B)
0.001
P(E)
0.002
B C E P(A)
T T T 0.98
T T F 0.94
T F T 0.96
T F F 0.95
F T T 0.45
F T F 0.25
F F T 0.29
F F F 0.001
A P(J)
T 0.90
F 0.05
A P(M)
T 0.70
F 0.01
If the introduced variable is highly erratic, it can invalidate even more of the CPT than we would like.
CrazyGuyP(C)
0.25
CS 460, Belief Networks 22
What if we cause a new variable?
Burglary Earthquake
Alarm
JohnCalls MaryCalls
P(B)
0.001
P(E)
0.002
B C E P(A)
T T T 0.98
T T F 0.94
T F T 0.96
T F F 0.95
F T T 0.45
F T F 0.25
F F T 0.29
F F F 0.001
A P(J)
T 0.90
F 0.05
A P(M)
T 0.70
F 0.01
However some changes to the CPT may be absurd so we may never have to worry about them.
CrazyGuy P(C)
0.25
CS 460, Belief Networks 23
Things in the model can change
We can account for change in the model over time (a more advanced topic). • John and Mary may be more or less likely to call at certain
times. Cyclical repetition may be not too difficult to model. • People may become tired of their job and be less likely call
over longer periods. This may be easy or difficult to model.• Crime picks up. If the trend is slow enough, the model may
be able to adjust online even if we have never observed crime picking up before. However, this may easily and totally throw our model off.• However, keep in mind, we may get good enough results for
our model even without accounting for changes over time.
CS 460, Belief Networks 24
How would we apply this to Robotics?
Tree in PathRock in
Path
Obstacle Detect
Execute Stop
ExecuteTurn
P(T)
0.25
P(R)
0.4
T R P(O)
T T 0.95
T F 0.45
F T 0.29
F F 0.001
O P(S)
T 0.90
F 0.05
O P(U)
T 0.70
F 0.01
The CPT can describe other events and probabilities such as action success given observations.
CS 460, Belief Networks 25
Exact Inference in a Bayesian Network
What if we want to make an inference such as: what is the probability of a tree in the path given that the robot has stopped and turned. This might be useful to a robot which can judge if there is a tree in a path based on the behavior of another robot. So if robot A sees robot B turn or stop it might infer that there is a tree in the path.
, ,r o
P t s u P t P r P o t r P s o P u o
, ,, ,
, , , ,
P t s u P t s uT s u
P t s u P t s u P t s u P t s u
P
Normalized we get:
CS 460, Belief Networks 26
Bayesian Inference cont’
0.9
0.9
0.7
0.
0.25
0.25
0
7
0.3
0.3
.25
0.6
0.4
0
0.45
0.95
0.5 0.6
0
.1
0.1.4
5
0.025 50.
P r
P r
P
P o t r
P o t r
P
P s o
P s o
P s o
P s o
P
o tr
P r
P t
P t
u o
P u o
P u o
P
r
P
P t
ut oo t rP
We compute P of tree and P of not tree and normalize. This is essentially an enumeration of all situations for tree and not tree.
P t
P t
P t
P r
P r
P r
P o t r
P o t r
P o t r
P s o
P s o
P
P u o
P u o
P u o
P o t r P u o
s o
P sPt orP
0.9
0.9
0.7
0.
0.75
0.75
0
7
0.3
0.3
.75
0.6
0.4
0.6
0.
0.001
0.29
0. 0.1
4 0.10.7
9 9
0.5
9
71
,P t s u
,P t s u
CS 460, Belief Networks 27
Bayesian Inference cont’
Finishing:
,
0.051975
0.00315
0.002025
0.00285
P t s u
,
0.0002835
0.05481
0.0134865
0.00639
P t s u
0.06
0.07497
0.06 0.07497,
0.06 0.07497 0.06 0.07497
0.4445,0.5555
, ,r o
P t s u P t P r P o t r P s o P u o
, ,, ,
, , , ,
P t s u P t s uT s u
P t s u P t s u P t s u P t s u
P
The P of a tree in the path is 0.4445
CS 460, Belief Networks 28
What about hidden variables?
CS 460, Belief Networks 29
Example: car diagnosis
CS 460, Belief Networks 30
Example: car insurance
CS 460, Belief Networks 31
Compact conditional distributions
CS 460, Belief Networks 32
Compact conditional distributions
Know
Infer
CS 460, Belief Networks 33
Hybrid (discrete+continuous) networks
If subsidy was a hidden variable would it be discrete and discreet?
CS 460, Belief Networks 34
Continuous child variables
CS 460, Belief Networks 35
Continuous child variables
CS 460, Belief Networks 36
Discrete variable w/ continuous parents
CS 460, Belief Networks 37
Discrete variable
CS 460, Belief Networks 38
Inference in belief networks
Exact inference by enumerationExact inference by variable eliminationApproximate inference by stochastic
simulationApproximate inference by Markov chain
Monte Carlo (MCMC)