CSC242: Intro to AI
Lecture 18
Quiz
Stop Time: 2:15
Exact Inference in
Bayesian Networks
P(X | e) =
=↵
P(X, e) =
n
XY
y i=1
X
P(X, e, y)
y
P (Xi | parents(Xi ))
Exact Inference in
Bayesian Networks
• Exact inference in BNs is NP-hard
• Can be shown to be as hard as computing
the number of satisfying assignments of a
propositional logic formula => #P-hard
Approximate Inference
in Bayesian Networks
The Goal
• Query variable X
• Evidence variables E , ..., E
• Observed values: e = < e , ..., e >
• Non-evidence, non-query (“hidden”) variables: Y
• Approximate: P(X | e)
1
m
1
m
A Simpler Goal
• Query variable X
• Evidence variables E , ..., E
• Observed values: e = < e , ..., e >
• Non-evidence, non-query (“hidden”) variables: Y
• Approximate: P(X)
1
m
1
m
A Really Simple Goal
Heads
Goal: P(Heads)
i.e., P(Heads=true)
i.e., P(heads)
A Really Simple Goal
Heads
# of flips:
# of heads:
N
Nheads
A Really Simple Goal
Heads
Nheads
P (heads) ⇡
N
A Really Simple Goal
Heads
Nheads
P (heads) = lim
N !1
N
Sampling
• Generating events (possible worlds) from a
distribution
• Estimating probabilities as ratio of observed
events to total events
• Consistent estimate: becomes exact in
the large-sample limit
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
R P(W)
t
.99
f
.90
t .90
f .00
C P(R)
t .80
f .20
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
R P(W)
t
.99
f
.90
t .90
f .00
P (Rain = true)
C P(R)
t .80
f .20
Sampling
• Generate assignments of values to the
random variables
• That is consistent with the full joint
distribution encoded in the network
• In the sense that in the limit, the
probability of any event is equal to the
frequency of its occurrence
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
R P(W)
t
.99
f
.90
t .90
f .00
hCloudy = true, Sprinkler = false, Rain = true, WetGrass = truei
Generating Samples
• Sample each variable in topological order
• Child appears after its parents
• Choose the value for that variable
conditioned on the values already chosen
for its parents
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
R P(W)
t
.99
f
.90
t .90
f .00
C P(R)
t .80
f .20
Cloudy
Sprinkler
Rain
WetGrass
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
Cloudy
Sprinkler
Rain
WetGrass
R P(W)
t
.99
f
.90
t .90
f .00
P(Cloudy) = h0.5, 0.5i
true
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
Cloudy
true
Sprinkler false
Rain
WetGrass
R P(W)
t
.99
f
.90
t .90
f .00
P(Sprinkler | Cloudy = true) = 0.1, 0.9⇥
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
Cloudy
true
Sprinkler false
Rain
true
WetGrass
R P(W)
t
.99
f
.90
t .90
f .00
P(Rain | Cloudy = true) = 0.8, 0.2⇥
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
Cloudy
true
Sprinkler false
Rain
true
WetGrass true
R P(W)
t
.99
f
.90
t .90
f .00
P(WetGrass | Sprinkler = false, Rain = true) = h0.9, 0.1i
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
Cloudy
true
Sprinkler false
Rain
true
WetGrass true
R P(W)
t
.99
f
.90
t .90
f .00
hCloudy = true, Sprinkler = false, Rain = true, WetGrass = truei
Guaranteed to be a consistent estimate
(becomes exact in the large-sample limit)
Sampling
• Generate an assignment of values to the
random variables
• That is consistent with the full joint
distribution encoded in the network
• In the sense that in the limit, the
probability of any event is equal to the
frequency of its occurrence
The Goal
• Query variable X
• Evidence variables E , ..., E
• Observed values: e = < e , ..., e >
• Non-evidence, non-query (“hidden”) variables: Y
• Approximate: P(X | e)
1
m
1
m
P(C)=.5
P(Rain | Sprinkler = true)
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
R P(W)
t
.99
f
.90
t .90
f .00
hCloudy = true, Sprinkler = false, Rain = true, WetGrass = truei
Rejection Sampling
• Generate sample from the prior
distribution specified by the network
• Reject sample if inconsistent with the
evidence
• Use remaining samples to estimate
probability of event
P(C)=.5
P(Rain | Sprinkler = true)
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
R P(W)
t
.99
f
.90
t .90
f .00
P(Rain | Sprinkler = true)
C P(R)
t .80
f .20
100 samples
Sprinkler=false: 73
Sprinkler=true: 27
Rain=true: 8
Rain=false: 19
8 19
⇥ , ⇤ = ⇥0.296, 0.704⇤
27 27
Rejection Sampling
• Generate sample from the prior
distribution specified by the network
• Reject sample if inconsistent with the
evidence
• Use remaining samples to estimate
probability of event
• Fraction of samples consistent with the
evidence drops exponentially with number
of evidence variables
Likelihood Weighting
• Generate only samples consistent with the
evidence
• i.e., fix values of evidence varables
• Instead of counting 1 for each non-rejected
sample, weight the count by the likelihood
(probability) of the sample given the
evidence
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
R P(W)
t
.99
f
.90
t .90
f .00
C P(R)
t .80
f .20
Cloudy
Sprinkler
Rain
WetGrass
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
Cloudy
Sprinkler
Rain
WetGrass
R P(W)
t
.99
f
.90
t .90
f .00
P(Rain | Cloudy = true, WetGrass = true)
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
R P(W)
t
.99
f
.90
t .90
f .00
C P(R)
t .80
f .20
Cloudy
Sprinkler
Rain
WetGrass
w = 1.0
P(Rain | Cloudy = true, WetGrass = true)
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
R P(W)
t
.99
f
.90
t .90
f .00
C P(R)
t .80
f .20
Cloudy
Sprinkler
Rain
WetGrass
w = 1.0
P(Rain | Cloudy = true, WetGrass = true)
true
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
R P(W)
t
.99
f
.90
t .90
f .00
C P(R)
t .80
f .20
Cloudy
true
Sprinkler false
Rain
WetGrass
w = 0.5
P(Rain | Cloudy = true, WetGrass = true)
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
R P(W)
t
.99
f
.90
t .90
f .00
C P(R)
t .80
f .20
Cloudy
true
Sprinkler false
Rain
true
WetGrass
w = 0.5
P(Rain | Cloudy = true, WetGrass = true)
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
R P(W)
t
.99
f
.90
t .90
f .00
C P(R)
t .80
f .20
Cloudy
true
Sprinkler false
Rain
true
WetGrass true
w = 0.5
P(Rain | Cloudy = true, WetGrass = true)
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
R P(W)
t
.99
f
.90
t .90
f .00
C P(R)
t .80
f .20
Cloudy
true
Sprinkler false
Rain
true
WetGrass true
w = 0.45
P(Rain | Cloudy = true, WetGrass = true)
Likelihood Weighting
• Generate sample using topological order
• Evidence variable: Fix value to evidence
value and update weight of sample using
probability in network
• Non-evidence variable: Sample from
values using probabilities in the network
(given parents)
Likelihood Weighting
• Pros:
• Doesn’t reject any samples
• Cons:
• More evidence lower weight
• Affected by order of evidence vars in
topsort (later = worse)
Approximate Inference
in Bayesian Networks
• Rejection Sampling
• Likelihood Weighting
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
Cloudy
true
Sprinkler true
Rain
false
WetGrass true
R P(W)
t
.99
f
.90
t .90
f .00
P(Rain | Sprinkler = true, WetGrass = true)
Markov Chain Monte
Carlo Simulation
• To approximate: P(X | e)
• Generate a sequence of states
• Values of evidence variables are fixed
• Values of other variables appear in the
right proportion given the distribution
encoded by the network
U1
...
Um
X
Z1j
Y1
...
U1
Z nj
Yn
Conditional
Independence
Um
...
X
Z 1j
Y1
...
Z nj
Yn
Markov
Blanket
Markov Blanket
• The Markov Blanket of a node is its
parents, its children, and its children’s
parents.
• A node is conditionally independent of all
other nodes in the network given its
Markov Blanket
MCMC
Gibbs Simulation
Sampling
• To approximate: P(X | e)
• Start in a state with evidence variables set
to evidence values (others arbitrary)
• On each step, sample the non-evidence
variables conditioned on the values of the
variables in their Markov Blankets
• Order irrelevant
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
Cloudy
true
Sprinkler true
Rain
false
WetGrass true
R P(W)
t
.99
f
.90
t .90
f .00
P(Rain | Sprinkler = true, WetGrass = true)
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
Cloudy
true
Sprinkler true
Rain
false
WetGrass true
R P(W)
t
.99
f
.90
t .90
f .00
P(Rain | Sprinkler = true, WetGrass = true)
P(Cloudy | Sprinkler = true, Rain = false)
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
Cloudy
false
Sprinkler true
Rain
false
WetGrass true
R P(W)
t
.99
f
.90
t .90
f .00
P(Rain | Sprinkler = true, WetGrass = true)
P(Rain | Sprinkler = true, Rain = false, Cloudy = false)
P(C)=.5
Cloudy
C
t
f
P(S)
.10
.50
Rain
Sprinkler
Wet
Grass
S
t
t
f
f
C P(R)
t .80
f .20
Cloudy
false
Sprinkler true
Rain
true
WetGrass true
R P(W)
t
.99
f
.90
t .90
f .00
P(Rain | Sprinkler = true, WetGrass = true)
Gibbs Sampling
• To approximate: P(X | e)
• Start in a state with evidence variables set
to evidence values (others arbitrary)
• On each step, sample non-evidence
variables conditioned on the values of the
variables in their Markov Blanket
• Order irrelevant
• A form of local search!
Exact Inference in
Bayesian Networks
• #P-Hard even for distribution described as
a Bayesian Network
Approximate Inference
in Bayesian Networks
• Sampling consistent with a distribution
• Rejection Sampling: rejects too much
• Likelihood Weighting: weights get too small
• Gibbs Sampling: MCMC algorithm (like local
search)
• All generate consistent estimates (equal to
exact probability in the large-sample limit)
Project 3
Inference in Bayesian Networks
Implement and evaluate
Full description:
http://www.cs.rochester.edu/courses/242/
spring2013/projects/project-03.pdf
Due: Fri 5 Apr 23:00
For Next Time:
AIMA 15.0-15.3