# Uncertainty

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```					Decision Trees
Bits
• We are watching a set of independent random samples of X
• We see that X has four possible values

• We transmit data over a binary serial link. We can encode
each reading with two bits (e.g. A=00, B=01, C=10, D = 11)

0100001001001110110011111100…
Fewer Bits
• Someone tells us that the probabilities are not equal

• It’s possible…
…to invent a coding for your transmission that only uses
1.75 bits on average per symbol. Here is one.
General Case
• Suppose X can have one of m values…

• What’s the smallest possible number of bits, on average, per
symbol, needed to transmit a stream of symbols drawn from X’s
distribution? It’s

entropy ( p1 ,..., pm )   p1 log 2 p1  ... pm log 2 pm

• H(X) : is the entropy of X
• Well, Shannon got to this formula by setting down several
desirable properties for uncertainty, and then finding it.
Constructing decision trees
• Normal procedure: top down in recursive divide-and-
conquer fashion
– First: an attribute is selected for root node and a branch is
created for each possible attribute value
– Then: the instances are split into subsets (one for each
branch extending from the node)
– Finally: the same procedure is repeated recursively for each
branch, using only instances that reach the branch
• Process stops if all instances have the same class
Which attribute to select?

(b)
(a)

(c)                  (d)
A criterion for attribute
•
selection
Which is the best attribute?

• The one which will result in the smallest tree
– Heuristic: choose the attribute that produces the “purest”
nodes

• Popular impurity criterion: entropy of nodes
– Lower the entropy purer the node.

• Strategy: choose attribute that results in lowest entropy of
the children nodes.
Example: attribute “Outlook”
Information gain
 Usually people don’t use directly the entropy of a node.
Rather the information gain is being used.

 Clearly, greater the information gain better the purity of a
node. So, we choose “Outlook” for the root.
Continuing to split
The final decision tree

• Note: not all leaves need to be pure; sometimes identical
instances have different classes
Splitting stops when data can’t be split any further
Highly-branching attributes
• The weather data with ID code
Tree stump for ID code
attribute
Highly-branching attributes
So,
• Subsets are more likely to be pure if there is a large
number of values
– Information gain is biased towards choosing attributes with a
large number of values
– This may result in overfitting (selection of an attribute that is
non-optimal for prediction)
The gain ratio
• Gain ratio: a modification of the information gain that
reduces its bias
• Gain ratio takes number and size of branches into account
when choosing an attribute
– It corrects the information gain by taking the intrinsic
information of a split into account
• Intrinsic information: entropy (with respect to the attribute
on focus) of node to be split.
Computing the gain ratio
Gain ratios for weather data
More on the gain ratio
• “Outlook” still comes out top but “Humidity” is now a much
closer contender because it splits the data into two

• However: “ID code” has still greater gain ratio. But its

• Problem with gain ratio: it may overcompensate
– May choose an attribute just because its intrinsic information
is very low
– Standard fix: choose an attribute that maximizes the gain
ratio, provided the information gain for that attribute is at
least as great as the average information gain for all the
attributes examined.
Discussion
• Algorithm for top-down induction of decision trees (“ID3”)
was developed by Ross Quinlan (University of Sydney
Australia)

• Gain ratio is just one modification of this basic algorithm
– Led to development of C4.5, which can deal with numeric
attributes, missing values, and noisy data

• There are many other attribute selection criteria! (But
almost no difference in accuracy of result.)

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 views: 2 posted: 10/11/2011 language: English pages: 19