CSC242: Intro to AI
Lecture 23
Learning: The Big
Picture So Far
Function Learning
Linear Regression
Linear Classifiers
Neural Networks
Learning Probabilistic
Models
100% cherry
75% cherry
25% lime
50% cherry
50% lime
25% cherry
75% lime
100% lime
D1=
D2=
Bags
Agent, process, disease, ...
Candies
Actions, effects, symptoms,
results of tests, ...
Observations
D3=
Goal
Predict next Predict agent’s next move
candy
Predict next output of process
Predict disease given symptoms
and tests
Learning and Bayesian
Networks
Learning and Bayesian
Networks
• The distribution defined by the network is
parameterized by the entries in the CPTs
associated with the nodes
• A BN defines a space of distributions
corresponding to the parameter space
Learning and Bayesian
Networks
• If we have a BN that we believe represents
the causality (conditional independence) in
our problem
• In order to find (estimate) the true
distribution...
• We need to learn the parameters of the
model from the training data
Burglary
JohnCalls
P(B)
.001
Earthquake
Alarm
B
t
t
f
f
A P(J)
t
f
.90
.05
E
t
f
t
f
P(E)
.002
P(A)
.95
.94
.29
.001
MaryCalls
A P(M)
t
f
.70
.01
hΘ
P(F=cherry)
Θ
Flavor
hΘ
P(F=cherry)
Θ
Flavor
N
c
l
Independent Identically
Distributed (i.i.d.)
• Probability of a sample is independent of
any previous samples
P(Di |Di
1 , Di 2 , . . .)
= P(Di )
• Probability distribution doesn’t change
among samples
P(Di ) = P(Di
1)
= P(Di
2)
= ···
P(F=cherry)
hΘ
Θ
Flavor
N
P (d | h⇥ ) =
=
c
l
Y
j
c
P (dj | h⇥ )
· (1
)l
Maximum Likelihood
Hypothesis
argmax P (d | h⇥ )
⇥
Maximum Likelihood
Hypothesis
argmax P (d | h⇥ )
⇥
Log Likelihood
P (d | h⇥ ) =
Y
=
c
L(d | h⇥ ) = log P (d | h⇥ ) =
j
P (dj | h⇥ )
· (1
X
j
)l
log P (dj | h⇥ )
= c log
+ l log(1
)
Maximum Likelihood
Hypothesis
L(d | h⇥ ) = c log
+ l log(1
c
c
argmax L(d | h⇥ ) =
=
c+l
N
⇥
)
Flavor
Wrapper
P(F=cherry)
Θ
Flavor
Wrapper
F P(W=red|F)
cherry
Θ1
lime
Θ2
h⇥,⇥1 ,⇥2
P(F=cherry)
Θ
Flavor
Wrapper
F P(W=red|F)
cherry
Θ1
lime
Θ2
P(F=cherry)
h⇥,⇥1 ,⇥2
Θ
Flavor
Wrapper
F P(W=red|F)
cherry
Θ1
lime
Θ2
P (F = f, W = w | h⇥,⇥1 ,⇥2 ) =
P (F = f | h⇥,⇥1 ,⇥2 ) · P (W = w | W = f, h⇥,⇥1 ,⇥2 )
P (F = c, W = g | h
,
1,
2
)=
· (1
1)
F
W
P(F=f,W=w| hΘ,Θ1,Θ2)
cherry
red
Θ Θ1
cherry
green
Θ (1-Θ1)
lime
red
(1-Θ) Θ2
lime
green
(1-Θ) (1-Θ2)
N
c
rc
l
gc
rl
gl
F
W
P
N=c+l
cherry
red
Θ Θ1
rc
cherry
green
Θ (1-Θ1)
gc
lime
red
(1-Θ) Θ2
rl
lime
green
(1-Θ) (1-Θ2)
gl
F
cherry
cherry
W
red
green
P
Θ Θ1
Θ (1-Θ1)
N=c+l P (d | h⇥,⇥1 ,⇥2 )
rc
gc
(
1)
( (1
rc
1 ))
lime
red
(1-Θ) Θ2
rl
((1
)
lime
green
(1-Θ) (1-Θ2)
rl
((1
)(1
gc
2)
rl
gl
))
2
P (d | h⇥,⇥1 ,⇥2 ) =
(
1)
=
c
rc
(1
· ( (1
l
) ·
1 ))
gc
rc
1 (1
L(d | h⇥,⇥1 ,⇥2 ) = c log
· ((1
1)
gc
)
·
2)
rl
rl
2 (1
+ l log(1
· ((1
gl
)
2
)+
[rc log
1
+ gc log(1
1 )]+
[rl log
2
+ gl log(1
2 )]
)(1
2 ))
gl
c
c
=
=
c+l
N
1
rc
rc
=
=
rc + g c
c
2
rl
rl
=
=
rl + g l
l
h⇥,⇥1 ,⇥2
P(F=cherry)
Θ
Flavor
Wrapper
F P(W=red|F)
cherry
Θ1
lime
Θ2
c
c
=
=
c+l
N
1
rc
rc
=
=
rc + g c
c
2
rl
rl
=
=
rl + g l
l
argmax L(d | h⇥,⇥1 ,⇥2 ) = argmax P (d | h⇥,⇥1 ,⇥2 )
⇥,⇥1 ,⇥2
⇥,⇥1 ,⇥2
Naive Bayes Models
Class
Attr1
Attr2
Attr3
...
Naive Bayes Models
{ mammal, reptile, fish, ... }
Class
Furry
Warm
Blooded
Size
...
Naive Bayes Models
Class
Attr1
Attr2
Attr3
...
Naive Bayes Models
{ mammal, reptile, fish, ... }
Class
Furry
Warm
Blooded
Size
...
Naive Bayes Models
{ terrorist, tourist }
Class
Arrival
Mode
One-way
Ticket
Furtive
Manner
...
Naive Bayes Models
Disease
Test1
Test2
Test3
...
Learning Naive Bayes
Models
• Naive Bayes model with n Boolean
attributes requires 2n+1 parameters
• Maximum likelihood hypothesis can be
found with no search
• Probabilities are observed frequencies
• Scales to large problems
• Robust to noisy or missing data
Learning with
Complete Data
• Can learn the CPTs for a Bayes Net from
observations that include values for all
variables
• Finding maximum likelihood parameters
decomposes into separate problems, one
for each parameter
• Parameter values for a variable given its
parents are the observed frequencies
{ terrorist, tourist }
Class
Arrival
Mode
One-way
Ticket
Furtive
Manner
...
Arrival One-Way Furtive
...
Class
taxi
yes
very
...
terrorist
car
no
none
...
tourist
car
yes
very
...
terrorist
car
yes
some
...
tourist
walk
yes
none
...
student
bus
no
some
...
tourist
Disease
Test1
Test2
Test3
...
Test
Test2
Test3
...
Disease
T
F
T
...
?
T
F
F
...
?
F
F
T
...
?
T
T
T
...
?
F
T
F
...
?
T
F
T
...
?
2
Smoking
2
Diet
2
Exercise
2
Smoking
2
Diet
2
Exercise
54 HeartDisease
6
Symptom 1
6
Symptom 2
6
Symptom 3
(a)
78 parameters
54
Symptom 1
162
Symptom 2
486
Symptom 3
(b)
708 parameters
Hidden (Latent)
Variables
• Can dramatically reduce the number of
parameters required to specify a Bayes net
• Reduces amount of data required to
learn the parameters
• Values of hidden variables not present in
training data (observations)
• “Complicates” the learning problem
EM
Expectation-Maximization
• Repeat
• Expectation: “Pretend” we know the
parameters and compute (or estimate)
likelihood of data given model
• Maximization: Recompute parameters
using expected values as if they were
observed values
• Until convergence
Learning: The Big
Picture for 242
Function Learning
Linear Regression
Linear Classifiers
Neural Networks
Learning Probabilistic Models (Bayes Nets)
Tue 23 Apr & Thu 25 Apr
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Get there early...