Overview of Course
So far,we have studied
The concept of Bayesian network
Independence and Separation in Bayesian networks
Inference in Bayesian networks
The rest of the course:Data analysis using Bayesian network
Parameter learning
:Learn parameters for a given structure.
Structure learning
:Learn both structures and parameters
Learning latent structures
:Discover latent variables behind observed
variables and determine their relationships.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 1/58
COMP538:Introduction to Bayesian Networks
Lecture 6:Parameter Learning in Bayesian Networks
Nevin L.Zhang
lzhang@cse.ust.hk
Department of Computer Science and Engineering
Hong Kong University of Science and Technology
Fall 2008
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 2/58
Objective
Objective:
Principles for parameter learning in Bayesian networks.
Algorithms for the case of complete data.
Reading:Zhang and Guo (2007),Chapter 7
Reference:Heckerman (1996) (ﬁrst half),Cowell et al (1999,Chapter 9)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 3/58
Problem Statement
Outline
1 Problem Statement
2 Principles of Parameter Learning
Maximum likelihood estimation
Bayesian estimation
Variable with Multiple Values
3 Parameter Estimation in General Bayesian Networks
The Parameters
Maximum likelihood estimation
Properties of MLE
Bayesian estimation
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 4/58
Problem Statement
Parameter Learning
Given:
A Bayesian network structure.
X
1
X
2
X
3
X
5
X
4
A data set
X
1
X
2
X
3
X
4
X
5
0
0
1
1
0
1
0
0
1
0
0
1
0
0
1
0
0
1
1
1
:
:
:
:
:
Estimate conditional probabilities:
P(X
1
),P(X
2
),P(X
3
X
1
,X
2
),P(X
4
X
1
),P(X
5
X
1
,X
3
,X
4
)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 5/58
Principles of Parameter Learning
Outline
1 Problem Statement
2 Principles of Parameter Learning
Maximum likelihood estimation
Bayesian estimation
Variable with Multiple Values
3 Parameter Estimation in General Bayesian Networks
The Parameters
Maximum likelihood estimation
Properties of MLE
Bayesian estimation
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 6/58
Principles of Parameter Learning
Maximum likelihood estimation
SingleNode Bayesian Network
T
H
X: result of tossing a thumbtack
X
Consider a Bayesian network with one
node X,where X is the result of tossing a
thumbtack and Ω
X
= {H,T}.
Data cases:
D
1
= H,D
2
= T,D
3
= H,...,D
m
= H
Data set:D = {D
1
,D
2
,D
3
,...,D
m
}
Estimate parameter:θ = P(X=H).
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 8/58
Principles of Parameter Learning
Maximum likelihood estimation
Likelihood
Data:D = {H,T,H,T,T,H,T}
As possible values of θ,which of the following is the most likely?Why?
θ = 0
θ = 0.01
θ = 10.5
θ = 0 contradicts data because P(Dθ = 0) = 0.It cannot explain the data
at all.
θ = 0.01 almost contradicts with the data.It does not explain the data well.
However,it is more consistent with the data than θ = 0 because
P(Dθ = 0.01) > P(Dθ = 0).
So θ = 0.5 is more consistent with the data than θ = 0.01 because
P(Dθ = 0.5) > P(Dθ = 0.01)
It explains the data the best among the three and is hence the most likely.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 9/58
Principles of Parameter Learning
Maximum likelihood estimation
Maximum Likelihood Estimation
*
L( D)
10
In general,the larger P(Dθ = v) is,the
more likely θ = v is.
Likelihood of parameter θ given data set:
L(θD) = P(Dθ)
The maximum likelihood estimation
(MLE) θ
∗
of θ is a possible value of θ
such that
L(θ
∗
D) = sup
θ
L(θD).
MLE best explains data or best ﬁts data.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 10/58
Principles of Parameter Learning
Maximum likelihood estimation
i.i.d and Likelihood
Assume the data cases D
1
,...,D
m
are independent given θ:
P(D
1
,...,D
m
θ) =
m
i =1
P(D
i
θ)
Assume the data cases are identically distributed:
P(D
i
= H) = θ,P(D
i
= T) = 1−θ for all i
(Note:
i.i.d means independent and identically distributed
)
Then
L(θD) = P(Dθ) = P(D
1
,...,D
m
θ)
=
m
i =1
P(D
i
θ) = θ
m
h
(1 −θ)
m
t
(1)
where m
h
is the number of heads and m
t
is the number of tail.
Binomial likelihood
.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 11/58
Principles of Parameter Learning
Maximum likelihood estimation
Example of Likelihood Function
Example:D = {D
1
= H,D
2
T,D
3
= H,D
4
= H,D
5
= T}
L(θD) = P(Dθ)
= P(D
1
= Hθ)P(D
2
= Tθ)P(D
3
= Hθ)P(D
4
= Hθ)P(D
5
= Tθ)
= θ(1 −θ)θθ(1 −θ)
= θ
3
(1 −θ)
2
.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 12/58
Principles of Parameter Learning
Maximum likelihood estimation
Suﬃcient Statistic
A
suﬃcient statistic
is a function s(D) of data that summarizing the
relevant information for computing the likelihood.That is
s(D) = s(D
) ⇒L(θD) = L(θD
)
Suﬃcient statistics tell us all there is to know about data.
Since L(θD) = θ
m
h
(1 −θ)
m
t
,
the pair (m
h
,m
t
) is a
suﬃcient statistic
.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 13/58
Principles of Parameter Learning
Maximum likelihood estimation
Loglikelihood
Loglikelihood:
l (θD) = logL(θD) = logθ
m
h
(1 −θ)
m
t
= m
h
logθ +m
t
log(1 −θ)
Maximizing likelihood is the same as maximizing loglikelihood.The latter is
easier.
By Corollary 1.1 of Lecture 1,the following value maximizes l (θD):
θ
∗
=
m
h
m
h
+m
t
=
m
h
m
MLE is intuitive.
It also has nice properties:
E.g.
Consistence
:θ
∗
approaches the true value of θ with probability 1
as m goes to inﬁnity.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 14/58
Principles of Parameter Learning
Bayesian estimation
Drawback of MLE
Thumbtack tossing:
(m
h
,m
t
) = (3,7).MLE:θ = 0.3.
Reasonable.Data suggest that the thumbtack is biased toward tail.
Coin tossing:
Case 1:(m
h
,m
t
) = (3,7).MLE:θ = 0.3.
Not reasonable.
Our experience (prior) suggests strongly that coins are fair,hence
θ=1/2.
The size of the data set is too small to convince us this particular coin
is biased.
The fact that we get (3,7) instead of (5,5) is probably due to
randomness.
Case 2:(m
h
,m
t
) = (30,000,70,000).MLE:θ = 0.3.
Reasonable.
Data suggest that the coin is after all biased,overshadowing our prior.
MLE does not diﬀerentiate between those two cases.It doe not take
prior information into account.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 16/58
Principles of Parameter Learning
Bayesian estimation
Two Views on Parameter Estimation
MLE
:
Assumes that θ is unknown but ﬁxed parameter.
Estimates it using θ
∗
,the value that maximizes the likelihood function
Makes prediction based on the estimation:P(D
m+1
= HD) = θ
∗
Bayesian Estimation
:
Treats θ as a random variable.
Assumes a prior probability of θ:p(θ)
Uses data to get posterior probability of θ:p(θD)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 17/58
Principles of Parameter Learning
Bayesian estimation
Two Views on Parameter Estimation
Bayesian Estimation
:
Predicting D
m+1
P(D
m+1
= HD) =
P(D
m+1
= H,θD)dθ
=
P(D
m+1
= Hθ,D)p(θD)dθ
=
P(D
m+1
= Hθ)p(θD)dθ
=
θp(θD)dθ.
Full Bayesian
:Take expectation over θ.
Bayesian MAP
:
P(D
m+1
= HD) = θ
∗
= arg max p(θD)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 18/58
Principles of Parameter Learning
Bayesian estimation
Calculating Bayesian Estimation
Posterior distribution:
p(θD) ∝ p(θ)L(θD)
= θ
m
h
(1 −θ)
m
t
p(θ)
where the equation follows from (1)
To facilitate analysis,assume prior has
Beta distribution
B(α
h
,α
t
)
p(θ) ∝ θ
α
h
−1
(1 −θ)
α
t
−1
Then
p(θD) ∝ θ
m
h
+α
h
−1
(1 −θ)
m
t
+α
t
−1
(2)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 19/58
Principles of Parameter Learning
Bayesian estimation
Beta Distribution
The normalization constant for
the Beta distribution B(α
h
,α
t
)
Γ(α
t
+α
h
)
Γ(α
t
)Γ(α
h
)
where Γ(.) is the Gamma
function.For any integer α,
Γ(α) = (α −1)!.It is also deﬁned
for nonintegers.
Density function of prior Beta
distribution B(α
h
,α
t
),
p(θ) =
Γ(α
t
+α
h
)
Γ(α
t
)Γ(α
h
)
θ
α
h
−1
(1 −θ)
α
t
−1
The
hyperparameters
α
h
and α
t
can be thought of as ”imaginary”
counts from our prior experiences.
Their sum α = α
h
+α
t
is called
equivalent sample size
.
The larger the equivalent sample
size,the more conﬁdent we are in
our prior.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 20/58
Principles of Parameter Learning
Bayesian estimation
Conjugate Families
Binomial Likelihood:θ
m
h
(1 −θ)
m
t
Beta Prior:θ
α
h
−1
(1 −θ)
α
t
−1
Beta Posterior:θ
m
h
+α
h
−1
(1 −θ)
m
t
+α
t
−1
.
Beta distributions are hence called a
conjugate family
for Binomial
likelihood.
Conjugate families allow closedform for posterior distribution of parameters
and closedform solution for prediction.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 21/58
Principles of Parameter Learning
Bayesian estimation
Calculating Prediction
We have
P(D
m+1
= HD) =
θp(θD)dθ
= c
θθ
m
h
+α
h
−1
(1 −θ)
m
t
+α
t
−1
dθ
=
m
h
+α
h
m+α
where c is the normalization constant,m=m
h
+m
t
,α = α
h
+α
t
.
Consequently,
P(D
m+1
= TD) =
m
t
+α
t
m+α
After taking data D into consideration,now our updated belief on X=T is
m
t
+α
t
m+α
.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 22/58
Principles of Parameter Learning
Bayesian estimation
MLE and Bayesian estimation
As m goes to inﬁnity,P(D
m+1
= HD) approaches the MLE
m
h
m
h
+m
t
,which
approaches the true value of θ with probability 1.
Coin tossing example revisited:
Suppose α
h
= α
t
= 100.Equivalent sample size:200
In case 1,
P(D
m+1
= HD) =
3 +100
10 +100 +100
≈ 0.5
Our prior prevails.
In case 2,
P(D
m+1
= HD) =
30,000 +100
100,0000 +100 +100
≈ 0.3
Data prevail.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 23/58
Principles of Parameter Learning
Variable with Multiple Values
Variable with Multiple Values
Bayesian networks with a single multivalued variable.
Ω
X
= {x
1
,x
2
,...,x
r
}.
Let θ
i
= P(X = x
i
) and θ = (θ
1
,θ
2
,...,θ
r
).
Note that θ
i
≥ 0 and
i
θ
i
= 1.
Suppose in a data set D,there are m
i
data cases where X takes value x
i
.
Then
L(θD) = P(Dθ) =
N
j=1
P(D
j
θ) =
r
i =1
θ
m
i
i
Multinomial likelihood.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 25/58
Principles of Parameter Learning
Variable with Multiple Values
Dirichlet distributions
Conjugate family for multinomial likelihood:
Dirichlet distributions
.
A Dirichlet distribution is parameterized by r parameters α
1
,α
2
,...,
α
r
.
Density function given by
Γ(α)
r
i =1
Γ(α
i
)
k
i =1
θ
α
i
−1
i
where α = α
1
+α
2
+...+α
r
.
Same as Beta distribution when r=2.
Fact:For any i:
θ
i
Γ(α)
r
i =1
Γ(α
i
)
k
i =1
θ
α
i
−1
i
dθ
1
dθ
2
...dθ
r
=
α
i
α
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 26/58
Principles of Parameter Learning
Variable with Multiple Values
Calculating Parameter Estimations
If the prior probability is a Dirichlet distribution Dir(α
1
,α
2
,...,α
r
),then
the posterior probability p(θD) is a given by
p(θD) ∝
r
i =1
θ
m
i
+α
i
−1
i
So it is Dirichlet distribution Dir(α
1
+m
1
,α
2
+m
2
,...,α
r
+m
r
),
Bayesian estimation has the following closedform:
P(D
m+1
=x
i
D) =
θ
i
p(θD)dθ =
α
i
+m
i
α +m
MLE:θ
∗
i
=
m
i
m
.(Exercise:Prove this.)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 27/58
Parameter Estimation in General Bayesian Networks
Outline
1 Problem Statement
2 Principles of Parameter Learning
Maximum likelihood estimation
Bayesian estimation
Variable with Multiple Values
3 Parameter Estimation in General Bayesian Networks
The Parameters
Maximum likelihood estimation
Properties of MLE
Bayesian estimation
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 28/58
Parameter Estimation in General Bayesian Networks
The Parameters
The Parameters
n variables:X
1
,X
2
,...,X
n
.
Number of states of X
i
:1,2,...,r
i
=Ω
X
i
.
Number of conﬁgurations of parents of X
i
:1,2,...,q
i
=Ω
pa(X
i
)
.
Parameters to be estimated:
θ
ijk
= P(X
i
= j pa(X
i
) = k),i = 1,...,n;j = 1,...,r
i
;k = 1,...,q
i
Parameter vector:θ = {θ
ijk
i = 1,...,n;j = 1,...,r
i
;k = 1,...,q
i
}.
Note that
j
θ
ijk
= 1∀i,k
θ
i..
:Vector of parameters for P(X
i
pa(X
i
))
θ
i..
= {θ
ijk
j = 1,...,r
i
;k = 1,...,q
i
}
θ
i.k
:Vector of parameters for P(X
i
pa(X
i
)=k)
θ
i.k
= {θ
ijk
j = 1,...,r
i
}
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 30/58
Parameter Estimation in General Bayesian Networks
The Parameters
The Parameters
Example:Consider the Bayesian network shown below.Assume all variables
are binary,taking values 1 and 2.
3
X
2
X
1
X
θ
111
= P(X
1
=1),θ
121
= P(X
1
=2)
θ
211
= P(X
2
=1),θ
221
= P(X
2
=2)
pa(X
3
) = 1:θ
311
= P(X
3
=1X
1
= 1,X
2
= 1),θ
321
= P(X
3
=2X
1
= 1,X
2
= 1)
pa(X
3
) = 2:θ
312
= P(X
3
=1X
1
= 1,X
2
= 2),θ
322
= P(X
3
=2X
1
= 1,X
2
= 2)
pa(X
3
) = 3:θ
313
= P(X
3
=1X
1
= 2,X
2
= 1),θ
323
= P(X
3
=2X
1
= 2,X
2
= 1)
pa(X
3
) = 4:θ
314
= P(X
3
=1X
1
= 2,X
2
= 2),θ
324
= P(X
3
=2X
1
= 2,X
2
= 2)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 31/58
Parameter Estimation in General Bayesian Networks
The Parameters
Data
A complete case D
l
:a vector of values,one for each variable.
Example:D
l
= (X
1
= 1,X
2
= 2,X
3
= 2)
Given:A set of complete cases:D = {D
1
,D
2
,...,D
m
}.
Example:
X
1
X
2
X
3
X
1
X
2
X
3
1 1 1
2 1 1
1 1 2
2 1 2
1 1 2
2 2 1
1 2 2
2 2 1
1 2 2
2 2 2
1 2 2
2 2 2
2 1 1
2 2 2
2 1 1
2 2 2
Find:The ML estimates of the parameters θ.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 32/58
Parameter Estimation in General Bayesian Networks
Maximum likelihood estimation
The Loglikelihood Function
Loglikelihood:
l (θD) = logL(θD) = logP(Dθ) = log
l
P(D
l
θ) =
l
logP(D
l
θ).
The term logP(D
l
θ):
D
4
= (1,2,2),
logP(D
4
θ) = logP(X
1
= 1,X
2
= 2,X
3
= 2)
= logP(X
1
=1θ)P(X
2
=2θ)P(X
3
=2X
1
=1,X
2
=2,θ)
= logθ
111
+logθ
221
+logθ
322
.
Recall:
θ = {θ
111
,θ
121
;θ
211
,θ
221
;θ
311
,θ
312
,θ
313
,θ
314
,θ
321
,θ
322
,θ
323
,θ
324
}
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 34/58
Parameter Estimation in General Bayesian Networks
Maximum likelihood estimation
The Loglikelihood Function
Deﬁne the
characteristic function
of case D
l
:
χ(i,j,k:D
l
) =
1 if X
i
= j,pa(X
i
) = k in D
l
0 otherwise
When l =4,D
4
= (1,2,2).
χ(1,1,1:D
4
) = χ(2,2,1:D
4
) = χ(3,2,2:D
4
) = 1
χ(i,j,k:D
4
) = 0 for all other i,j,k
So,logP(D
4
θ) =
ijk
χ(i,j,k;D
4
)logθ
ijk
In general,
logP(D
l
θ) =
ijk
χ(i,j,k:D
l
)logθ
ijk
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 35/58
Parameter Estimation in General Bayesian Networks
Maximum likelihood estimation
The Loglikelihood Function
Deﬁne
m
ijk
=
l
χ(i,j,k:D
l
).
It is the number of data cases where X
i
= j and pa(X
i
) = k.
Then
l (θD) =
l
logP(D
l
θ)
=
l
i,j,k
χ(i,j,k:D
l
)logθ
ijk
=
i,j,k
l
χ(i,j,k:D
l
)logθ
ijk
=
ijk
m
ijk
logθ
ijk
=
i,k
j
m
ijk
logθ
ijk
.(4)
Suﬃcient statistics:m
ijk
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 36/58
Parameter Estimation in General Bayesian Networks
Maximum likelihood estimation
MLE
Want:
arg max
θ
l (θD) = arg max
θ
ijk
i,k
j
m
ijk
logθ
ijk
Note that θ
ijk
= P(X
i
=j pa(X
i
)=k) and θ
i
j
k
= P(X
i
=j
pa(X
i
)=k
) are
not related if either i =i
or k=k
.
Consequently,we can separately maximize each term in the summation
i,k
[...]
arg max
θ
ijk
j
m
ijk
logθ
ijk
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 37/58
Parameter Estimation in General Bayesian Networks
Maximum likelihood estimation
MLE
By Corollary 1.1,we get
θ
∗
ijk
=
m
ijk
j
m
ijk
In words,the MLE estimate for θ
ijk
= P(X
i
=j pa(X
i
)=k) is:
θ
∗
ijk
=
number of cases where X
i
=j and pa(X
i
)=k
number of cases where pa(X
i
)=k
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 38/58
Parameter Estimation in General Bayesian Networks
Maximum likelihood estimation
Example
Example:
3
X
2
X
1
X
X
1
X
2
X
3
X
1
X
2
X
3
1 1 1
2 1 1
1 1 2
2 1 2
1 1 2
2 2 1
1 2 2
2 2 1
1 2 2
2 2 2
1 2 2
2 2 2
2 1 1
2 2 2
2 1 1
2 2 2
MLE for P(X
1
=1) is:6/16
MLE for P(X
2
=1) is:7/16
MLE for P(X
3
=1X
1
=2,X
2
=2) is:2/6
...
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 39/58
Parameter Estimation in General Bayesian Networks
Properties of MLE
A Question
Start from a joint distribution P(X) (Generative Distribution)
D:collection of data sampled from P(X).
Let S be a BN structrue (DAG) over variables X.
Learn parameters θ
∗
for BN structure S from D.
Let P
∗
(X) be the joint probability of the BN (S,θ
∗
).
Note:θ
∗
ijk
= P
∗
(X
i
=j pa
S
(X
i
)=k)
How is P
∗
related to P?
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 41/58
Parameter Estimation in General Bayesian Networks
Properties of MLE
MLE in General Bayesian Networks with Complete Data
.
to S
factorize according
Distributions that
*
P
P
*
P
∗
factorizes according to S.
P does not necessarily factorize
according to S.
We will show that,with
probability 1,P
∗
converges to the
distribution that
Factorizes according to S,
Is closest to P under KL
divergence among all
distributions that factorize
according to S.
If P factorizes according to S,P
∗
converges to P with probability 1.
(MLE is consistent.)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 42/58
Parameter Estimation in General Bayesian Networks
Properties of MLE
The Target Distribution
Deﬁne
θ
S
ijk
= P(X
i
=j pa
S
(X
i
) = k))
Let P
S
(X) be the joint distribution of the BN (S,θ
S
)
P
S
factorizes according to S and for any X ∈ X,
P
S
(Xpa(X)) = P(Xpa(X))
If P factorizes according to S,then P and P
S
are identical.
If P does not factorize according to S,then P and P
S
are diﬀerent.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 43/58
Parameter Estimation in General Bayesian Networks
Properties of MLE
First Theorem
Theorem (6.1)
Among all distributions Q that factorizes according to S,the KL divergence
KL(P,Q) is minimized by Q=P
S
.
P
S
is the closest to P among all those that factorize according to S.
Proof:
Since
KL(P,Q) =
X
P(X)log
P(X)
Q(X)
It suﬃces to show that
Proposition:Q=P
S
maximizes
X
P(X)logQ(X)
We show the claim by induction on the number of nodes.
When there is only one node,the proposition follows from property of KL
divergence (Corollary 1.1).
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 44/58
Parameter Estimation in General Bayesian Networks
Properties of MLE
First Theorem
Suppose the proposition is true for the case of n nodes.Consider the case of
n+1 nodes.
Let X be a leaf node and X
=X\{X}.S
be the obtained from S by
removing X.
Then
X
P(X)logQ(X) =
X
P(X
)logQ(X
) +
pa(X)
P(pa(X))
X
P(Xpa(X))logQ(Xpa(X))
By the induction hypothesis,the ﬁrst term is maximized by P
S
.
By Corollary 1.1,the second term is maximized if
Q(Xpa(X)) = P(Xpa(X)).
Hence the sum is maximized by P
S
.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 45/58
Parameter Estimation in General Bayesian Networks
Properties of MLE
Second Theorem
Theorem (6.2)
lim
N→∞
P
∗
(X=x) = P
S
(X=x) with probability 1
where N is the sample size,i.e.number of cases in D.
Proof:
Let
ˆ
P(X) be the
empirical distribution
:
ˆ
P(X=x) = fraction of cases in D where X=x
It is clear that
P
∗
(X
i
=j pa
S
(X
i
)=k) = θ
∗
ijk
=
ˆ
P(X
i
=j pa
S
(X
i
)=k)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 46/58
Parameter Estimation in General Bayesian Networks
Properties of MLE
Second Theorem
On the other hand,by the law of large numbers,we have
lim
N→∞
ˆ
P(X=x) = P(X=x) with probability 1
Hence
lim
N→∞
P
∗
(X
i
=j pa
S
(X
i
)=k) = lim
N→∞
ˆ
P(X
i
=j pa
S
(X
i
)=k)
= P(X
i
=j pa
S
(X
i
)=k) with probability 1
= P
S
(X
i
=j pa
S
(X
i
)=k)
Because both P
∗
and P
S
factorizes according to S,the theorem follows.
Q.E.D.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 47/58
Parameter Estimation in General Bayesian Networks
Properties of MLE
A Corollary
Corollary
If P factorizes according to S,then
lim
N→∞
P
∗
(X=x) = P(X=x) with probability 1
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 48/58
Parameter Estimation in General Bayesian Networks
Bayesian estimation
Bayesian Estimation
View θ as a vector of random variables with prior distribution p(θ).
Posterior:
p(θD) ∝ p(θ)L(θD)
= p(θ)
i,k
j
θ
m
ijk
ijk
where the equation follows from (4).
Assumptions need to be made about prior distribution.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 50/58
Parameter Estimation in General Bayesian Networks
Bayesian estimation
Assumptions
Global independence
in prior distribution:
p(θ) =
i
p(θ
i..
)
Local independence
in prior distribution:For each i
p(θ
i..
) =
k
p(θ
i.k
)
Parameter independence
= global independence + local independence:
p(θ) =
i,k
p(θ
i.k
)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 51/58
Parameter Estimation in General Bayesian Networks
Bayesian estimation
Assumptions
Further assume that p(θ
i.k
) is Dirichlet distribution Dir(α
i 0k
,α
i 1k
,...,α
ir
i
k
):
p(θ
i.k
) ∝
j
θ
α
ijk
−1
ijk
Then,
p(θ) =
i,k
j
θ
α
ijk
−1
ijk
product Dirichlet distribution
.
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 52/58
Parameter Estimation in General Bayesian Networks
Bayesian estimation
Bayesian Estimation
Posterior:
p(θD) ∝ p(θ)
i,k
j
θ
m
ijk
ijk
= [
i,k
j
θ
α
ijk
−1
ijk
]
i,k
j
θ
m
ijk
ijk
=
i,k
j
θ
m
ijk
+α
ijk
−1
ijk
It is also a product product Dirichlet distribution.(Think:What does this
mean?)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 53/58
Parameter Estimation in General Bayesian Networks
Bayesian estimation
Prediction
Predicting D
m+1
= {X
m+1
1
,X
m+1
2
,...,X
m+1
n
}.Random variables.
For notational simplicity,simply write D
m+1
= {X
1
,X
2
,...,X
n
}.
First,we have:
P(D
m+1
D) = P(X
1
,X
2
,...,X
n
D) =
i
P(X
i
pa(X
i
),D)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 54/58
Parameter Estimation in General Bayesian Networks
Bayesian estimation
Proof
P(D
m+1
D) =
P(D
m+1
θ)p(θD)dθ
P(D
m+1
θ) = P(X
1
,X
2
,...,X
n
θ)
=
i
P(X
i
pa(X
i
),θ)
=
i
P(X
i
pa(X
i
),θ
i..
)
p(θ
i
D) =
i
p(θ
i..
D)
Hence
P(D
m+1
D) =
i
P(X
i
pa(X
i
),θ
i..
)p(θ
i..
D)dθ
i..
=
i
P(X
i
pa(X
i
),D)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 55/58
Parameter Estimation in General Bayesian Networks
Bayesian estimation
Prediction
Further,we have
P(X
i
=j pa(X
i
)=k,D) =
P(X
i
=j pa(X
i
) = k,θ
ijk
)p(θ
ijk
D)dθ
ijk
=
θ
ijk
p(θ
ijk
D)dθ
ijk
Because
p(θ
i.k
D) ∝
j
θ
m
ijk
+α
ijk
−1
ijk
We have
θ
ijk
p(θ
ijk
D)dθ
ijk
=
m
ijk
+α
ijk
j
(m
ijk
+α
ijk
)
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 56/58
Parameter Estimation in General Bayesian Networks
Bayesian estimation
Prediction
Conclusion:
P(X
1
,X
2
,...,X
n
D) =
i
P(X
i
pa(X
i
),D)
where
P(X
i
=j pa(X
i
)=k,D) =
m
ijk
+α
ijk
m
i ∗k
+α
i ∗k
where m
i ∗k
=
j
m
ijk
and α
i ∗k
=
j
α
ijk
Notes:
Conditional independence or structure preserved after absorbing D.
Important property for sequential learning where we process one case
at a time.
The ﬁnal result is independent of the order by which cases are
processed.
Comparison with MLE estimation:
θ
∗
ijk
=
m
ijk
j
m
ijk
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 57/58
Parameter Estimation in General Bayesian Networks
Bayesian estimation
Summary
θ:random variable.
Prior p(θ):product Dirichlet distribution
p(θ) =
i,k
p(θ
i.k
) ∝
i,k
j
θ
α
ijk
−1
ijk
Posterior p(θD):also product Dirichlet distribution
p(θD) ∝
i,k
j
θ
m
ijk
+α
ijk
−1
ijk
Prediction:
P(D
m+1
D) = P(X
1
,X
2
,...,X
n
D) =
i
P(X
i
pa(X
i
),D)
where
P(X
i
=j pa(X
i
)=k,D) =
m
ijk
+α
ijk
m
i ∗k
+α
i ∗k
Nevin L.Zhang (HKUST)
Bayesian Networks
Fall 2008 58/58
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