Non-Informative Dirichlet Score for learning Bayesian networks

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Non-Informative Dirichlet Score
for learning Bayesian networks
Maomi Ueno and Masaki Uto
University of ElectroCommunications, Japan
BDe(Bayesian Dirichlet equivalence)
Heckerman, Geiger and Chickering (1995)
Non-informative prior : BDeu
As Buntine (1991) described  ijk   /(ri qi )
is considered to be a special case of the
BDe metric. Heckerman et al. (1995)
called this special case “BDeu”.
gh
Problems of BDeu
BDeu is the most popular Bayesian network
score.
However, recent reports have described that
learning Bayesian network is highly
sensitive to the chosen equivalent sample
size (ESS) in the Bayesian Dirichlet
equivalence uniform (BDeu) and this often
engenders some unstable or undesirable
results.
(Steck and Jaakkola, 2002, Silander,
Kontkanen and Myllymaki, 2007).
Theorem 1(Ueno 2010) When α is sufficiently
large, the log-ML is approximated
asymptotically as
1 N qi ri ri  1  nijk 
log p( X | g )  H ( g ,  )  H ( g ,  , X )  
log 1 
 O(1).


2 i 1 j 1 k 1 ri
  ijk 
Log-posterior
The number of
Parameters
The difference between
prior and data
As the two structures become equivalent, the
penalty term is minimized with the fixed ESS.
Conversely, as the two structures become
different, the term increases.
Moreover, α
determines the magnitude of the user‘s prior
belief for a hypothetical structure. Thus, the
mechanism of BDe makes sense to us when we
have approximate prior knowledge.
BDeu is not non-informative!!
Ueno (2011) showed that the prior of BDeu does
not represent ignorance of prior knowledge, but
rather a user’s prior belief in the uniformity of the
conditional distribution.
The results further imply that the optimal ESS
becomes large/small when the empirical
conditional distribution becomes uniform/skewed.
This main factor underpins the sensitivity of BDeu
to ESS.
Marginalization of ESS α
• Silander, Kontkanen and Myllymaki (2007)
proposed a learning method to marginalize
the ESS of BDeu. They averaged BDeu values
with increasing ESS from 1 to 100 by 1.0 .
• Cano et al. (2011) proposed averaging a wide
range of different chosen ESSs: ESS > 1:0, ESS
>> 1:0, ESS < 1:0, ESS << 1:0.
Why do we use Bayesian approach ?
The previous works eliminate out the ESS.
However,
One of most important features of
Bayesian approach is using prior
knowledge to augment the data.
→
When the score has a non-informative
prior, the ESS is expected to work
effectively as actual pseudo-samples to
augment the data.
Main idea
To obtain a non-informative prior,
we assume all possible hypothetical structures
as a prior belief of BDe because a problem of
BDeu is that it assumes only a uniform
distribution as a prior belief.
NIP-BDe
Definition 1. NIP-BDe (Non Informative Prior Bayesian Dirichlet
equivalence) is defined as
p( X |  , g ) 
h
p
(
g
)
p
(
X
|

,
g
,
g
)

g h G
Calculation method
Our method has the following steps:
1. Estimate the conditional probability parameters set
g
 g  { ijk
}, (i  1,  , N , j  1,  , qi , k  1,  , ri  1)
h
h
given gh from data.

p
2. Estimate the joint probability ( xi  k ,  g 
First, we estimate the conditional probability
parameters set given gh as
i

gh
ijk
j | g, g )
1
gh
 nijk
gh
rq
 i i
,
h
1
g

n
ij
gh
qi
Next, we calculate the estimated joint probability
as shown below.

p( xi  k ,  gi  j | g h ,  g ) 
h

p( x1 , , xi , , xN | g h , g )
h
xl x i   g
i
h
Expected log-BDe
In practice, however, the Expected log-Bde is
difficult to calculate because the product of
multiple probabilities suffers serious
computational problems. To avoid this, we
propose an alternative method, Expected logBDe, as described below.
Eg h G log p( X |  , g , g h ) 
h
p
(
g
)
log
p
(
X
|

,
g
,
g
)

g h G
Advantage
The optimal ESS of BDeu becomes large/small when
the empirical conditional distribution becomes
uniform/skewed because its hypothetical
structure assumes a uniform conditional
probabilities distribution and the ESS adjusts the
magnitude of the user’s belief for a hypothetical
structure.
However, the ESS of full non-informative prior is
expected to work effectively as actual pseudosamples to augment the data, especially when the
sample size is small, regardless of the uniformity
of empirical distribution.
Learning algorithm using dynamic programming
We employ the learning algorithm of (Silander and Myllymaki, 2006)
using dynamic programming. Our algorithm comprises four steps:
1. Compute the local Expected log-Bde scores for all possible
N 2 N 1 ( xi ,  g ) pairs.
2. For each variable xi  x , find the best parent set in parent
candidate set { g } for all  g  x / xi ,
3. Find the best sink for all 2N variables and the best ordering.
4. Find the optimal Bayesian network
Only Step 1 is different from the procedure described by Silander et
al., 2006. Algorithm 1 gives a pseudo-code for computing the local
Expected log-BDe scores.
i
i
i
Algorithm
Computational costs
• The time complexity of the Algorithm 1 is
O(N222(N-1)exp(w)) given an elimination order
of width w. The traditional methods (BDeu,
AIC, BIC, and so on) run in O(N22(N-1)).
• However, the required memory for the
proposed computation is equal to that of the
traditional scores.
Experiments
Conclusions
This paper presented a proposal for a
noninformative Dirichlet score by
marginalizing the possible hypothetical
structures as the user’s prior belief.
The results suggest that the proposed method is
effective especially for a small sample size.

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