Bayesian Networks
4
th
, December 2009
Presented by
Kwak
, Nam

ju
The slides are based on
<Data Mining : Practical Learning Tools and Techniques>, 2nd ed.,
written by Ian H. Witten &
Eibe
Frank.
Images and Materials are from the official lecture slides of the book.
Table of Contents
•
Probability Estimate vs. Prediction
•
What is Bayesian Network?
•
A Simple Example
•
A Complex One
•
Why does it work?
•
Learning Bayesian Networks
•
Overfitting
•
Searching for a Good Network Structure
•
K2 Algorithm
•
Other Algorithms
•
Conditional Likelihood
•
Data Structures for Fast Learning
Probability Estimate vs. Prediction
•
Naïve
Bayes
classifier, logistic regression
models: probability estimates
•
For each class, they estimate the probability
that a given instance belongs to that class.
Probability Estimate vs. Prediction
•
Why probability estimates are useful?
–
They allow predictions to be ranked.
–
Treat classification learning as the task of
learning class probability estimates from the data.
•
What is being estimated is
–
The conditional probability distribution of the
values of the class attribute given the values of
the other attributes.
Probability Estimate vs. Prediction
•
In this way, Naïve
Bayes
classifiers, logistic
regression models and decision trees are
ways of representing a conditional probability
distribution.
What is Bayesian Network?
•
A theoretically well

founded way of
representing probability distributions
concisely and comprehensively in a
graphical manner.
•
They are drawn as a
network of nodes
, one
for each attribute, connected by
directed
edges
in such a way that there are
no cycles
.
–
A directed acyclic graph
A Simple Example
Pr[outlook=rainy  play=no]
Summed up into 1
A Complex One
•
When outlook=rainy,
temperature=cool,
humidity=high, and
windy=true…
•
Let’s call E the situation
given above.
A Complex One
•
E: rainy, cool, high, and true
•
Pr[play=no, E] =
0.0025
•
Pr[play=yes, E] =
0.0077
Multiply all those!!
An additional example of the calculation
A Complex One
•
E: rainy, cool, high, and true
•
Pr[play=no, E] = 0.0025
•
Pr[play=yes, E] = 0.0077
A Complex One
Summed up into 1
Why does it work?
•
Terminologies
–
T: all the nodes, P: parents, D: descendant
–
Non

descendant: T

D
Non

descendants
Why does it work?
•
Assumption (conditional independence)
–
Pr[node  parents plus any other set of non

descendants]
= Pr[node  parents]
•
Chain rule
•
The nodes are ordered to give all ancestors of a
node
ai
indices smaller than
i
. It’s possible since the
network is acyclic.
Why does it work?
Ok, that’s what I’m talking about!!!
Learning Bayesian Networks
•
Basic components of algorithms for learning
Bayesian networks:
–
Methods for evaluating the goodness of a given
network
–
Methods for searching through space of possible
networks
Learning Bayesian Networks
•
Methods for evaluating the goodness of a given
network
–
Calculate the probability that the network accords
to each instance and multiply these probabilities
all together.
–
Alternatively, use the sum of logarithms.
•
Methods for searching through space of possible
networks
–
Search through the space of possible sets of
edges.
Overfitting
•
While maximizing the log

likelihood based on the
training data, the resulting network may
overfit
. What
are the solutions?
–
Cross

validation: training instances and validation
instances (similar to ‘early stopping’ in learning of
neural networks)
–
Penalty for the complexity of the network
–
Assign a prior distribution over network structures
and find the most likely network using the
probability by the data.
Overfitting
•
Penalty for the complexity of the network
–
Based on the total # of independent estimates in
all the probability tables, which is called the # of
parameters
Overfitting
•
Penalty for the complexity of the network
–
K: the # of parameters
–
LL: log

likelihood
–
N: the # of instances in the training data
–
AIC score =

LL+K
–
MDL score =

LL+(K/2)
logN
–
Those two scores are supposed to be minimized.
Akaike
Information
Criterion
Minimum
Description Length
Overfitting
•
Assign a prior distribution over network
structures and find the most likely network by
combining its prior probability with the
probability accorded to the network by the
data.
Searching for
a Good Network Structure
•
The probability of a single instance is the
product of all the individual probabilities from
the various conditional probability tables.
•
The product can be rewritten to group
together all factors relating to the same table.
•
Log

likelihood can also be grouped in such a
way.
Searching for
a Good Network Structure
•
Therefore log

likelihood can be optimized
separately for each node.
•
This can be done by adding, or removing
edges from other nodes to the node being
optimized. (without making cycles)
Which one is the best?
Searching for
a Good Network Structure
•
AIC and MDL can be dealt with in a similar
way since they can be split into several
components, one for each node.
K2 Algorithm
•
Starts with given ordering of
nodes
(attributes
)
•
Processes
each node in turn
•
Greedily
tries adding edges
from previous
nodes to current node
•
Moves
to next node when current
node can’t
be optimized
further
Result depends on
the initial order
K2 Algorithm
•
Some tricks
–
Use Naïve
Bayes
classifier as a starting point.
–
Ensure that every node is in the
Marcov
blanket
of the class node. (
Marcov
blanket: parents,
children, and children’s parents)
Naïve Bayesian Classifier
Marcov
blanket
Pictures from Wikipedia and
http://nltk.googlecode.com/svn/trunk/doc/book/ch06.html
Other Algorithms
•
Extended K2
–
sophisticated but slow
–
Do not order the nodes.
–
Greedily add or delete edges between arbitrary
pairs of nodes.
•
Tree Augmented Naïve
Bayes
(TAN)
Other Algorithms
•
Tree Augmented Naïve
Bayes
(TAN)
–
Augment a tree to a Naïve
Bayes
classifier.
–
When the class node and its outgoing edges are
eliminated, the remaining edges should form a
tree.
Naïve
Bayes
classifier
Tree
Pictures from
http://www.usenix.org/events/osdi04/tech/full_papers
/cohen/cohen_html/index.html
Other Algorithms
•
Tree Augmented Naïve
Bayes
(TAN)
–
MST of the network will be a clue for maximizing
likelihood.
Conditional Likelihood
•
What we actually need to know is the
conditional likelihood, which is the
conditional probability of the class given the
other attributes.
•
However, what we’ve tried to maximize is, in
fact, just the likelihood.
O
X
Conditional Likelihood
•
Computing the conditional likelihood for a
given network and dataset is straightforward.
•
This is what logistic regression does.
Data Structures for Fast Learning
•
Learning Bayesian networks involves a lot of
counting.
•
For each network structure to be searched,
the data must be scanned to get the
conditional probability tables.
(Since the
‘given term’ of the table of a certain node
changes frequently, we should rescan the
data in order to get the brand new
conditional probabilities many times.)
Data Structures for Fast Learning
•
Use a general hash tables.
–
Assuming that there are 5 attributes, 2 with 3
values and 3 with 2 values.
–
There’re 4*4*3*3*3=432 possible categories.
–
This calculation includes cases of missing values.
(i.e. null)
–
This can cause memory problems.
Data Structures for Fast Learning
•
AD (all

dimensions) tree
–
Using a general hash table, there will be
3*3*3=27 categories, even though only 8
categories are actually used.
Data Structures for Fast Learning
•
AD (all

dimensions) tree
Only 8 categories are required,
compared to 27.
Data Structures for Fast Learning
•
AD (all

dimensions) tree

construction
–
Assume each attribute in the data has
been assigned an
index.
–
Then
, expand node for attribute
i
with
the
values
of all
attributes
j >
i
–
Two
important restrictions:
•
Most
populous expansion for each attribute
is omitted
(breaking ties arbitrarily)
•
Expansions
with counts that are zero are
also omitted
–
The
root node is given index zero
Data Structures for Fast Learning
•
AD (all

dimensions) tree
Data Structures for Fast Learning
•
AD (all

dimensions) tree
Q. # of (humidity=normal, windy=true, play=no)?
Data Structures for Fast Learning
•
AD (all

dimensions) tree
Q. # of (humidity=normal, windy=false, play=no)?
?
Data Structures for Fast Learning
•
AD (all

dimensions) tree
Q. # of (humidity=normal, windy=false, play=no)?
#(humidity=normal, play=no)
–
#(humidity=normal, windy=true, play=no)
= 1

1=0
Data Structures for Fast Learning
•
AD tree only pay off if the data contains
many thousands of instances.
Questions and Answers
•
Any question?
Pictures from
http://news.ninemsn.com.au/article.aspx?id=805150
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