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Evaluation of Learning Models
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Literature:
T. Mitchel, Machine Learning, chapter 5
I.H. Witten and E. Frank, Data Mining, chapter
5
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Fayyad’s KDD Methodology
Transformed
data
Target
data
Patterns
Processed
data
Data Mining
data
Selection
Knowledge
Interpretation
Evaluation
Transformation
Preprocessing & feature
selection
& cleaning
2
Overview of the lecture
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Evaluating Hypotheses (errors, accuracy)
Comparing Hypotheses
Comparing Learning Algorithms (hold-out
methods)
Performance Measures
Varia (Occam’s razor, warning)
No Free Lunch
3
Evaluating Hypotheses:
Two definitions of error

True error of hypothesis h with respect to
target function f and distribution D is the
probability that h will misclassify an instance
drawn at random according to D.
errorD (h)  Pr [ f ( x)  h( x)]
xD
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Two definitions of error (2)

Sample error of hypothesis h with respect to
target function f and data sample S is the
proportion of examples h misclassifies
errorS (h) 
1
n
  ( f ( x)  h( x))
xS
where  ( f ( x)  h( x)) is 1 if
and 0 otherwise.
f ( x )  h( x )
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Two definitions of error (3)

How well does errorS(h) estimate errorD(h)?
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Problems estimating error
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1. Bias: If S is training set, errorS(h) is
optimistically biased
bias  E[errorS (h)]  errorD (h)

For unbiased estimate, h and S must be
chosen independently.
2. Variance: Even with unbiased S, errorS(h)
may still vary from errorD(h).
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Example
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Hypothesis h misclassifies 12 of the 40
examples in S
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errorS (h) 
 0.30
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What is errorD(h)?
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Estimators
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Experiment:
1. choose sample S of size n according to
distribution D
2. measure errorS(h)
errorS(h) is a random variable (i.e., result of
an experiment)
errorS(h) is an unbiased estimator for
errorD(h)
Given observed errorS(h) what can we
conclude about errorD(h)?
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Confidence intervals

If
 S contains n examples, drawn independently of h
and each other
 n  30
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Then
 With approximately 95% probability, errorD(h) lies
in the interval
errorS (h)  1.96
errorS ( h )(1errorS ( h ))
n
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Confidence intervals (2)

If
 S contains n examples, drawn independently of h
and each other
 n  30

Then
 With approximately N% probability, errorD(h) lies in
the interval
errorS (h)  z N
errorS ( h )(1errorS ( h ))
n
where
N%: 50% 68% 80% 90% 95% 98% 99%
zN:
0.67 1.00 1.28 1.64 1.96 2.33 2.58
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errorS(h) is a random variable


Rerun the experiment with different randomly
drawn S (of size n)
Probability of observing r misclassified
examples:
n!
r
nr
P(r ) 
errorD (h) (1  errorD (h))
r!(n  r )!
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Binomial probability distribution

Probability P(r) of r heads in n coin flips,
if p = Pr(heads)
Binomial distribution
for n = 10 and p = 0.3
n!
r
nr
P(r ) 
errorD (h) (1  errorD (h))
r!(n  r )!
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Binomial probability distribution (2)
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Expected, or mean value of X, E[X], is
n
E[ X ]   iP (i )  np
i 0
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Variance of X is
Var ( X )  E[( X  E[ X ]) ]  np(1  p)
2
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Standard deviation of X, X, is
 X  E[( X  E[ X ]) ]  np(1  p)
2
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Normal distribution approximates
binomial

errorS(h) follows a Binomial distribution, with
 mean  error ( h )  errorD ( h)
S
 standard deviation
 error ( h) 
S

errorD ( h )(1errorD ( h ))
n
Approximate this by a Normal distribution with
 mean  errorS ( h )  errorD ( h)
 standard deviation
 error ( h) 
S
errorD ( h )(1errorD ( h ))
n
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Normal probability distribution
p ( x) 

1
2
a


e
The probability that X will fall into the interval
b
(a,b) is given by


2
 12 ( x ) 2
p ( x)dx
Expected, or mean value of X, E[X], is
E[X] = 
Variance of X is Var(x) = 2
Standard deviation of X, X , X = 
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Normal probability distribution
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80% of area (probability) lies in   1.28
N% of area (probability) lies in   zN
N%:
zN:
50% 68% 80% 90% 95% 98% 99%
0.67 1.00 1.28 1.64 1.96 2.33 2.58
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Confidence intervals, more correctly

If
 S contains n examples, drawn independently of h
and each other
 n  30

Then
 with approximately 95% probability, errorS(h) lies in
the interval
errorD (h)  1.96
errorD ( h )(1errorD ( h ))
n
errorS (h)  1.96
errorS ( h )(1errorS ( h ))
n
 and errorD(h) approximately lies in the interval
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Central Limit Theorem

Consider a set of independent, identically
distributed random variables Y1...Yn, all
governed by an arbitrary probability
distribution with mean  and finite variance 2.
Define the sample mean,
n
Y 

1
n
Y
i 1
i
Central Limit Theorem. As n  , the
distribution governing Y approaches a Normal
distribution, with mean  and variance 2/n.
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Comparing Hypotheses:
Difference between hypotheses
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Test h1 on sample S1, test h2 on S2
1. Pick parameter to estimate
d  errorD (h1 )  errorD (h2 )
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2. Choose an estimator
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3. Determine probability distribution that
governs estimator
dˆ  errorS1 (h1 )  errorS2 (h2 )
 dˆ 
errorS1 ( h1 )(1errorS1 ( h1 ))
n1

errorS2 ( h2 )(1errorS2 ( h2 ))
n2
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Difference between hypotheses (2)
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4. Find interval (L, U) such that N% of
probability mass falls in the interval
dˆ  z N
errorS1 ( h1 )(1errorS1 ( h1 ))
n1

errorS2 ( h2 )(1errorS2 ( h2 ))
n2
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Paired t test to compare hA, hB
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1. Partition data into k disjoint test sets
T1,T2,…,Tk of equal size, where this size is at
least 30.
2. For i from 1 to k, do
 i  errorT (hA )  errorT (hB )
 3. Return the value  , where
i
 
i
k
1
k

i 0
i
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Paired t test to compare hA, hB (2)
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N% confidence interval estimate for d:
 
s 

k
1
k

i 0
i
k
1
k ( k 1)
 (
i 0
i
 )
2
Note  i approximately Normally distributed
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Comparing learning algorithms:
LA and LB
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What we’d like to estimate:
ES  D [errorD ( LA (S ))  errorD ( LB (S ))]
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where L(S) is the hypothesis output by learner
L using training set S
I.e., the expected difference in true error
between hypotheses output by learners LA and
LB, when trained using randomly selected
training sets S drawn according to distribution
D.
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Comparing learning algorithms LA
and LB (2)
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But, given limited data D0, what is a good
estimator?
 Could partition D0 into training set S0 and test set
T0, and measure
errorT0 ( LA ( S 0 ))  errorT0 ( LB ( S 0 ))
 Even better, repeat this many times and average
the results
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Comparing learning algorithms LA
and LB (3): k-fold cross validation
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1. Partition data D0 into k disjoint test sets
T1,T2,…,Tk of equal size, where this size is at
least 30.
2. For i from 1 to k, do
use Ti for the test set, and the remaining
data for training set Si
3. Return the average of the errors on the test
sets
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Practical Aspects
A note on parameter tuning
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It is important that the test data is not used in any
way to create the classifier
Some learning schemes operate in two stages:
 Stage 1: builds the basic structure
 Stage 2: optimizes parameter settings
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The test data can’t be used for parameter tuning!
Proper procedure uses three sets: training data,
validation data, and test data
Validation data is used to optimize parameters
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Holdout estimation, stratification
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What shall we do if the amount of data is limited?
The holdout method reserves a certain amount for
testing and uses the remainder for training
 Usually: one third for testing, the rest for training
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Problem: the samples might not be representative
 Example: class might be missing in the test data
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Advanced version uses stratification
 Ensures that each class is represented with approximately
equal proportions in both subsets
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More on cross-validation
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Standard method for evaluation: stratified ten-fold
cross-validation
Why ten? Extensive experiments have shown that this
is the best choice to get an accurate estimate
 There is also some theoretical evidence for this
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Stratification reduces the estimate’s variance
Even better: repeated stratified cross-validation
 E.g. ten-fold cross-validation is repeated ten times
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and results are averaged (reduces the variance)
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Estimation of the accuracy of a
learning algorithm
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10-fold cross validation gives a pessimistic estimate of
the accuracy of the hypothesis build on all training
data, provided that the law “the more training data
the better” holds.
For model selection 10-fold cross validation often
works fine.
An other method is: leave-one-out or jackknife (N-fold
cross validation with N = training set size).
Also the standard deviation is essential for comparing
learning algorithms.
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Performance Measures:
Issues in evaluation
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Statistical reliability of estimated differences in
performance
Choice of performance measure
 Number of correct classifications
 Accuracy of probability estimates
 Error in numeric predictions
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Costs assigned to different types of errors
 Many practical applications involve costs
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Counting the costs
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In practice, different types of classification
errors often incur different costs
Examples:
 Predicting when cows are in heat (“in estrus”)
 “Not in estrus” correct 97% of the time
 Loan decisions
 Oil-slick detection
 Fault diagnosis
 Promotional mailing
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Taking costs into account
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The confusion matrix:
predicted class
yes
actual
class
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no
yes True positive
False negative
no
True negative
False positive
There are many other types of costs!
 E.g.: costs of collecting training data
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Lift charts
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In practice, costs are rarely known
Decisions are usually made by comparing
possible scenarios
Example: promotional mailout
 Situation 1: classifier predicts that 0.1% of all
households will respond
 Situation 2: classifier predicts that 0.4% of the
10000 most promising households will respond
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A lift chart allows for a visual comparison
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Generating a lift chart
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Instances are sorted according to their predicted
probability of being a true positive:
Rank
1
2
3
4
…
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Predicted probability Actual class
0.95
Yes
0.93
Yes
0.93
No
0.88
Yes
…
…
In lift chart, x axis is sample size and y axis is number
of true positives
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A hypothetical lift chart
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Probabilities, reliability
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In order to generate a lift chart we need
information that tells that one classification is
more probable/reliable to be of the class of
interest than another classification.
The Naïve Bayes classifier and also Nearest
Neighbor classifiers can output such
information.
If we start to fill our subset with the most
probable examples, the subset will contain a
larger proportion desired elements.
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Summary of measures
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Varia:
Model selection criteria
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Model selection criteria attempt to find a good
compromise between:
A. The complexity of a model
B. Its prediction accuracy on the training data
Reasoning: a good model is a simple model
that achieves high accuracy on the given data
Also known as Occam’s Razor: the best theory
is the smallest one that describes all the facts
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Warning
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Suppose you are gathering hypotheses that
have a probability of 95% to have an error
level below 10%
What if you have found 100 hypotheses
satisfying this condition?
Then the probability that all have an error
below 10% is equal to (0.95)100  0.013
corresponding to 1.3 %. So, the probability of
having at least one hypothesis with an error
above 10% is about 98.7%!
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No Free Lunch!!
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Theorem (no free lunch)
 For any two learning algorithms LA and LB the
following is true, independently of the sampling
distribution and the number of training instances n:
 Uniformly averaged over all target functions F,
E(errorS(hA) | F, n) = E(errorS(hB) | F, n)
 Idem for a fixed training set D
 See: Mitchell ch.2
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No Free Lunch (2)

Sketch of proof:
 If all functions are possible, for each function F for which LA
outperforms LB a function F’ can be found with the opposite
conclusion.
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Conclusion: Without assumptions on the underlying
functions, the hypothesis space, no generalization is
possible!
In order to generalize, machine learning algorithms
need some kind of bias:
 inductive bias : not all possible functions are in the hypothesis
space (restriction bias) or not all possible functions will be
found because of the search strategy (preference or search
bias) (Mitchell, ch. 2, 3).
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No free lunch (3)
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The other way around: for any ML algorithm
there exist data sets on which it performs well
and there exist data sets on which it performs
badly!
We hope that the latter sets do not occur too
often in real life.
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Summary of notions
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True error, sample error
Bias of sample error
Accuracy, confidence intervals
Central limit theorem
Paired t test
k-fold cross validation, leave-one-out, holdout
Stratification
Training set, validation set, and test set
Confusion matrix, TP, FP, TN, FN
Lift chart, ROC curve, Recall-precision curve
Occam’s razor
No free lunch
Inductive bias; restriction bias; search bias
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