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概率模型
Review:probability(example)
Randomvariables:sunshineS∈{0,1},rainR∈{0,1}
Jointdistribution:
srP(S=s,R=r)
0
1
0
1
0
P(S,R)=0
1
1
Marginaldistribution:
sP(S=s)
P(S)=
1

(aggregaterows)
1

Conditionaldistribution:
sP(S=s|R=1)
P(S|R=1)=

(selectrows,normalize)
Review:probability(general)
Randomvariables:
X=(X1,...,Xn)partitionedinto(A,B)
Jointdistribution:
b
P(X)=P(X1,...,Xn)
Marginaldistribution:
P(A)=ΣP(A,B=b)
Conditionaldistribution:
P(A|B=b)∝P(A,B=b)
Probabilisticinferencetask
Randomvariables:unknownquantitiesintheworld
X=(S,R,T,A)
Inwords:
•Observeevidence(trafficinautumn):T=1,A=1
•Interestedinquery(rain?):R
Insymbols:
16
Challenges
Modeling:HowtospecifyajointdistributionP(X1,...,Xn)com-
pactly?
Bayesiannetworks(factorgraphsforprobabilitydistributions)
Inference:HowtocomputequeriesP(R|T=1,A=1)efficiently?
Variableelimination,Gibbssampling,particlefiltering(analogueof
algorithmsforfindingmaximumweightassignment)
Bayesiannetwork(alarm)
P(B=b,E=e,A=a)=p(b)p(e)p(a|b,e)
B
A
E
bp(b)
ep(e)
beap(a|b,e)
1s
01−s
1s
01−s
0001
0010
0100
0111
1000
1011
1100
1111
p(b)=s·[b=1]+(1−s)·[b=0]
p(e)=s·[e=1]+(1−s)·[e=0]
p(a|b,e)=[a=(b∨e)]
Probabilisticinference(alarm)
Jointdistribution:
b
0
0
0
0
1
1
1
1
e
0
0
1
1
0
0
1
1
a
0
1
0
1
0
1
0
1
P(B=b,E=e,A=a)
(1−s)2
0
0
(1−s)s
0
s(1−s)
0
s2
Queries:P(B)?P(B|A=1)?P(B|A=1,E=1)?
Explainingaway
B
E
A
Keyidea:explainingaway

theeffect,conditioningononecausereducestheprobabilityofthe
othercause.
P(X1=x1,...,Xn=xn)=
Definition
def
Definition:Bayesiannetwork
LetX=(X1,...,Xn)berandomvariables.
ABayesiannetworkisadirectedacyclicgraph(DAG)thatspec-
ifiesajointdistributionoverXasaproductoflocalconditional
distributions,oneforeachnode:
n
Πp(xi|xParents(i))
i=1
Specialproperties
Keydifferencefromgeneralfactorgraphs:
Keyidea:locallynormalized
Implications:
•Consistencyofsub-Bayesiannetworks
•Consistencyofconditionaldistributions