b ayesian n etworks. s ome a pplications of bn medical diagnosis troubleshooting of...
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BAYESIAN NETWORKS
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SOME APPLICATIONS OF BN
Medical diagnosis Troubleshooting of hardware/software
systems Fraud/uncollectible debt detection Data mining Analysis of genetic sequences Data interpretation, computer vision, image
understanding
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MORE COMPLICATED SINGLY-CONNECTED BELIEF NET
Radio
Battery
SparkPlugs
Starts
Gas
Moves
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Region = {Sky, Tree, Grass, Rock}
R2
R4R3
R1
Above
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CALCULATION OF JOINT PROBABILITY
B E P(A|…)
TTFF
TFTF
0.950.940.290.001
Burglary Earthquake
Alarm
MaryCallsJohnCalls
P(B)
0.001
P(E)
0.002
A P(J|…)
TF
0.900.05
A P(M|…)
TF
0.700.01
P(JMABE)= P(J|A)P(M|A)P(A|B,E)P(B)P(E)= 0.9 x 0.7 x 0.001 x 0.999 x 0.998= 0.00062
P(x1x2…xn) = Pi=1,…,nP(xi|parents(Xi))
full joint distribution table
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WHAT DOES THE BN ENCODE?
Burglary EarthquakeJohnCalls MaryCalls | AlarmJohnCalls Burglary | AlarmJohnCalls Earthquake | AlarmMaryCalls Burglary | AlarmMaryCalls Earthquake | Alarm
Burglary Earthquake
Alarm
MaryCallsJohnCalls
A node is independent of its non-descendents, given its parents
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PROBABILISTIC INFERENCE
Is the following problem…. Given:
A belief state P(X1,…,Xn) in some form (e.g., a Bayes net or a joint probability table)
A query variable indexed by q Some subset of evidence variables indexed by
e1,…,ek
Find: P(Xq | Xe1 ,…, Xek)
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B E P(A|…)
TTFF
TFTF
0.950.940.290.001
Burglary Earthquake
Alarm
MaryCallsJohnCalls
P(B)
0.001
P(E)
0.002
A P(J|…)
TF
0.900.05
A P(M|…)
TF
0.700.01
TOP-DOWN INFERENCE: RECURSIVE COMPUTATION OF ALL MARGINALS DOWNSTREAM OF EVIDENCE
P(A|E) = P(A|B,E)P(B) +P(A| B,E)P(B)
P(J|E) = P(J|A,E)P(A) +P(J| A,E)P(A) P(M|E) = P(M|A,E)P(A) +
P(M| A,E)P(A)
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TOP-DOWN INFERENCE
Only works if the graph of ancestors of a variable is a polytree
Evidence given on ancestor(s) of the query variable
Efficient: O(d 2k) time, where d is the number of ancestors
of a variable, with k a bound on # of parents Evidence on an ancestor cuts off influence of
portion of graph above evidence node
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QUERYING THE BN
The BN gives P(T|C) P(C|T) can be computed using
Bayes rule:
P(A|B) = P(B|A) P(A) / P(B)
Cavity
Toothache
P(C)
0.1
C P(T|C)
TF
0.40.01111
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QUERYING THE BN The BN gives P(T|C) What about P(C|T)? P(Cavity|Toothache) =
P(Toothache|Cavity) P(Cavity)
P(Toothache)
[Bayes’ rule]
Querying a BN is just applying Bayes’ rule on a larger scale…
Cavity
Toothache
P(C)
0.1
C P(T|C)
TF
0.40.01111 Denominator computed by
summing out numerator over Cavity and Cavity
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NAÏVE BAYES MODELS
P(Cause,Effect1,…,Effectn)= P(Cause) Pi P(Effecti | Cause)
Cause
Effect1 Effect2 Effectn
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NAÏVE BAYES CLASSIFIER
P(Class,Feature1,…,Featuren)= P(Class) Pi P(Featurei | Class)
Class
Feature1 Feature2 Featuren
P(C|F1,….,Fk) = P(C,F1,….,Fk)/P(F1,….,Fk)
= 1/Z P(C) Pi P(Fi|C)
Given features, what class?
Spam / Not Spam
English / French/ Latin
…
Word occurrences
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COMMENTS ON NAÏVE BAYES MODELS
Very scalable (thousands or millions of features!), easy to implement
Easily handles missing data: just ignore the feature
Conditional independence of features is main weakness. What if two features were actually correlated? Many features?
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VARIABLE ELIMINATION: PROBABILISTIC INFERENCE IN GENERAL NETWORKS
Coherence
Difficulty Intelligence
Happy
Grade SAT
Letter
Job
Basic idea: Eliminate “nuisance” variables one
at a time via marginalization
Example: P(J)
Elimination order: C,D,I,H,G,S,L
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Coherence
Difficulty Intelligence
Happy
Grade SAT
Letter
Job
P(D|C)
P(C)
P(I)
P(G|I,D)
P(H|G,J)
P(J|S,L)
P(S|I)
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Coherence
Difficulty Intelligence
Happy
Grade SAT
Letter
Job
P(D|C)
P(C)
ELIMINATING C
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Difficulty Intelligence
Happy
Grade SAT
Letter
Job
P(D)=cP(D|C)P(C)
C IS ELIMINATED, GIVING A NEW FACTOR OVER D
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Difficulty Intelligence
Happy
Grade SAT
Letter
Job
P(D)
ELIMINATING D
P(G|I,D)
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Intelligence
Happy
Grade SAT
Letter
Job
D IS ELIMINATED, GIVING A NEW FACTOR OVER G, I
P(G|I)=dP(G|I,d)P(d)
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Intelligence
Happy
Grade SAT
Letter
Job
ELIMINATING I
P(G|I) P(S|I)
P(I)
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Happy
Grade SAT
Letter
Job
I IS ELIMINATED, PRODUCING A NEW FILL EDGE AND FACTOR OVER G AND S
P(G,S)=iP(i)P(G|i)P(S|i)
New undirected fill edge
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Happy
Grade SAT
Letter
Job
ELIMINATING H
P(H|G,J)
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Happy
Grade SAT
Letter
Job
ELIMINATING H
P(H|G,J)
fGJ(G,J)=hP(h|G,J)=1
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Grade SAT
Letter
Job
H IS ELIMINATED, PRODUCING A NEW FILL EDGE AND FACTOR OVER G, J
fGJ(G,J)
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Grade SAT
Letter
Job
ELIMINATING G
fGJ(G,J)
P(G,S)
P(L|G)
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Grade SAT
Letter
Job
G IS ELIMINATED, MAKING A NEW TRINARY FACTOR OVER S,L,J AND A NEW FILL EDGE
fGJ(G,J)
P(G,S)
P(L|G)
fSLJ(S,L,J) = g P(g,S) P(L|g) fGJ(g,J)
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SAT
Letter
Job
ELIMINATING S
fSLJ(S,L,J)
P(J|S,L)
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SAT
Letter
Job
S IS ELIMINATED, CREATING A NEW FACTOR OVER L, J
fSLJ(S,L,J)
P(J|S,L)
fLJ(L,J) = s fSLJ(s,L,J) P(J|s ,L)
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Letter
Job
ELIMINATING L
fLJ(L,J)
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Letter
Job
L IS ELIMINATED, GIVING A NEW FACTOR OVER J (WHICH TURNS OUT TO BE P(J))
fLJ(L,J)
P(J)=l fLJ(l,J)
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Job
L IS ELIMINATED, GIVING A NEW FACTOR OVER J (WHICH TURNS OUT TO BE P(J))
P(J)
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JOINT DISTRIBUTION
P(X) = P(C)P(D|C)P(I)P(G|I,D)P(S|I)P(L|G) P(J|L,S)P(H|G,J)
Apply elimination ordering C,D,I,H,G,S,L
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GOING THROUGH VE
P(X) = P(C)P(D|C)P(I)P(G|I,D)P(S|I)P(L|G) P(J|L,S)P(H|G,J)
Apply elimination ordering C,D,I,H,G,S,L
fD(D)=SCP(C)P(D|C)
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GOING THROUGH VE
SCP(X) = fD(D)P(I)P(G|I,D)P(S|I)P(L|G) P(J|L,S)P(H|G,J)
Apply elimination ordering C,D,I,H,G,S,L
fD(D)=SCP(C)P(D|C)
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GOING THROUGH VE
SCP(X) = fD(D)P(I)P(G|I,D)P(S|I)P(L|G) P(J|L,S)P(H|G,J)
Apply elimination ordering C,D,I,H,G,S,L
fGI(G,I)=SDfD(D)P(G|I,D)
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GOING THROUGH VE
SC,DP(X) = fGI(G,I)P(I)P(S|I)P(L|G) P(J|L,S)P(H|G,J)
Apply elimination ordering C,D,I,H,G,S,L
fGI(G,I)=SDfD(D)P(G|I,D)
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GOING THROUGH VE
SC,DP(X) = fGI(G,I)P(I)P(S|I)P(L|G) P(J|L,S)P(H|G,J)
Apply elimination ordering C,D,I,H,G,S,L
fGS(G,S)=SIfGI(G,I)P(I)P(S|I)
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GOING THROUGH VE
SC,D,IP(X) = fGS(G,S)P(L|G)P(J|L,S)P(H|G,J) Apply elimination ordering C,D,I,H,G,S,L
fGS(G,S)=SIfGI(G,I)P(I)P(S|I)
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GOING THROUGH VE
SC,D,IP(X) = fGS(G,S)P(L|G)P(J|L,S)P(H|G,J) Apply elimination ordering C,D,I,H,G,S,L
fGJ(G,J)=SHP(H|G,J)
What values does this factor store?
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GOING THROUGH VE
SC,D,I,HP(X) = fGS(G,S)P(L|G)P(J|L,S)fGJ(G,J) Apply elimination ordering C,D,I,H,G,S,L
fGJ(G,J)=SHP(H|G,J)
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GOING THROUGH VE
SC,D,I,HP(X) = fGS(G,S)P(L|G)P(J|L,S)fGJ(G,J) Apply elimination ordering C,D,I,H,G,S,L
fSLJ(S,L,J)=SG fGS(G,S)P(L|G)fGJ(G,J)
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GOING THROUGH VE
SC,D,I,H,GP(X) = fSLJ(S,L,J)P(J|L,S) Apply elimination ordering C,D,I,H,G,S,L
fSLJ(S,L,J)=SG fGS(G,S)P(L|G)fGJ(G,J)
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GOING THROUGH VE
SC,D,I,H,GP(X) = fSLJ(S,L,J)P(J|L,S) Apply elimination ordering C,D,I,H,G,S,L
fLJ(L,J)=SS fSLJ(S,L,J)P(J|L,S)
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GOING THROUGH VE
SC,D,I,H,G,SP(X) = fLJ(L,J) Apply elimination ordering C,D,I,H,G,S,L
fLJ(L,J)=SS fSLJ(S,L,J)
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GOING THROUGH VE
SC,D,I,H,G,SP(X) = fLJ(L,J) Apply elimination ordering C,D,I,H,G,S,L
fJ(J)=SL fLJ(L,J)
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GOING THROUGH VE
SC,D,I,H,G,S,LP(X) = fJ(J) Apply elimination ordering C,D,I,H,G,S,L
fJ(J)=SL fLJ(L,J)
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ORDER-DEPENDENCE
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ORDER MATTERS
Coherence
Difficulty Intelligence
Happy
Grade SAT
Letter
Job
If we were to eliminate G first, we’d create a factor over D, I, L, and H (their distribution becomes coupled)
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ELIMINATION ORDER MATTERS
Coherence
Difficulty Intelligence
Happy
SAT
Letter
Job
If we were to eliminate G first, we’d create a factor over D, I, L, and H (their distribution becomes coupled)
fDILH(D,I,L,H) = g P(g|D,I)*P(L|g)*P(H|g)
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COMPLEXITY
In polytree networks where each node has at most k parents, O(n2k) with top-down ordering
In other networks, intermediate factors may involve more than k terms Worst case O(n) Good ordering heuristics exist, e.g. min-
neighbors, min-fill
Exact inference on non-polytree networks is NP-hard!
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VARIABLE ELIMINATION WITH EVIDENCE
Coherence
Difficulty Intelligence
Happy
Grade SAT
Letter
Job
Two-step process:1. Find P(X,e) with VE2. Normalize by P(e)
Example: P(J|H)
1. Run VE, enforcing H=T when H is eliminated.
2. This produces P(J,H=T) (a factor over J)
3. P(J=T|H=T) = P(J=T,H=T) / (P(J=T,H=T)+P(J=F,H=T))
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RECAP
Exact inference techniques Top-down inference: linear time when
ancestors of query variable are polytree, evidence is on ancestors
Bottom-up inference in Naïve Bayes models General inference using Variable Elimination
(We’ll come back to approximation techniques in a week.)
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NEXT TIME
Learning Bayes nets R&N 20.1-2