instance based and bayesian learning
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Instance based and Bayesian learning
Kurt Driessens
with slide ideas from a.o. Hendrik Blockeel, Pedro Domingos, David Page, Tom Dietterich and Eamon Keogh
Overview
Nearest neighbor methods– Similarity– Problems: • dimensionality of data, efficiency, etc.
– Solutions:• weighting, edited NN, kD-trees, etc.
Naïve Bayes– Including an introduction to Bayesian ML methods
Nearest Neighbor: A very simple idea
Imagine the world’s music collection represented in some space
When you like a song, other songs residing close to it should also be interesting …
Picture from Oracle
Nearest Neighbor Algorithm
1. Store all the examples <xi,yi>
2. Classify a new example x by finding the stored example xk that most resembles it and predicts that example’s class yk
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Some properties
• Learning is very fast (although we come back to this later)
• No information is lost• Hypothesis space– variable size– complexity of the hypothesis rises with the
number of stored examples
Decision Boundaries
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Voronoi diagram
Boundaries are not
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Keeping All Information
Advantage: no details lost Disadvantage: "details" may be noise
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k-Nearest-Neighbor: kNN
To improve robustness against noisy learning examples, use a set of nearest neighbors
For classification: use voting
k-Nearest-Neighbor: kNN (2)
For regression: use the mean
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Lazy vs Eager Learning
kNN doesn’t do anything until it needs to make a prediction = lazy learner– Learning is fast!– Predictions require work and can be slow
Eager learners start computing as soon as they receive dataDecision tree algorithms, neural networks, …– Learning can be slow– Predictions are usually fast!
Similarity measures
Distance metrics: measure of dis-similarityE.g. Manhattan, Euclidean or Ln-norm for numerical
attributes
Hamming distance for nominal attributes
Distance definition = critical!
E.g. comparing humans1. 1.85m, 37yrs2. 1.83m, 35yrs3. 1.65m, 37yrs
d(1,2) = 2.00…0999975…
d(1,3) = 0.2d(2,3) = 2.00808…
1. 185cm, 37yrs2. 183cm, 35yrs3. 165cm, 37yrs
d(1,2) = 2.8284…
d(1,3) = 20.0997…
d(2,3) = 18.1107…
Normalize attribute values
Rescale all dimensions such that the range is equal, e.g. [-1,1] or [0,1]
For [0,1] range:
with mi the minimum and Mi the maximum value for attribute i
Curse of dimensionality
Assume a uniformly distributed set of 5000 examples
To capture 5 nearest neighbors we need:– in 1 dim: 0.1% of the range– in 2 dim: = 3.1% of the range– in n dim: 0.1%1/n
Curse of Dimensionality (2)
With 5000 points in 10 dimensions, each attribute range must be covered approx. 50% to find 5 neighbors …
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Curse of Noisy Features
Irrelevant features destroy the metric’s meaningfulnessConsider a 1dim problem where the query x is at the origin,
the nearest neighbor x1 is at 0.1 and the second neighbor x2 at 0.5 (after normalization)
Now add a uniformly random feature. What is the probability that x2 becomes the closest neighbor?
approx. 15% !!
Curse of Noisy Features (2)
Location of x1 vs x2 on informative dimension
Weighted Distances
Solution: Give each attribute a different weight in the distance computation
for each attribute
for each class
for each example in
that class
Selecting attribute weights
Several options:– Experimentally find out which weights work well
(cross-validation)– Other solutions, e.g. (Langley,1996)
1. Normalize attributes (to scale 0-1)2. Then select weights according to "average attribute
similarity within class”
More distances
Strings– Levenshtein distance/edit distance= minimal number of changes needed to change one
word into the otherAllowed edits/changes:1.delete character2.insert character3.change character
(not used by some other edit-distances)
Even more distances
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D(Q,C)
Given two time series: Q = q1…qn
C = c1…cn
Euclidean
Start and end times are critical!
D(Q,R)
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Sequence distances (2)
Fixed Time AxisSequences are aligned “one to one”.
“Warped” Time AxisNonlinear alignments are possible.
edit
distance!
Distance-weighted kNN
k places arbitrary border on example relevance– Idea: give higher weight to closer instances
Can now use all training instances instead of only k (“Shepard’s method”)
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! In high-dimensional spaces, a function of d that “goes to zero fast enough” is needed. (Again “curse of dimensionality”.)
Fast Learning – Slow Predictions
Efficiency– For each prediction, kNN needs to compute the
distance (i.e. compare all attributes) for ALL stored examples
– Prediction time = linear in the size of the data-set
For large training sets and/or complex distances, this can be too slow to be practical
(1) Edited k-nearest neighbor
Use only part of the training dataIncremental deletion
of examples
Incremental
addition of
examples
✔ Less storage✗Order dependent✗ Sensitive to noisy data
More advanced alternatives exist (= IB3)
(2) Pipeline filters
Reduce time spent on far-away examples by using more efficient distance-estimates first
– Eliminate most examples using rough distance approximations
– Compute more precise distances for examples in the neighborhood
(3) kD-trees
Use a clever data-structure to eliminate the need to compute all distances
kD-trees are similar to decision trees except– splits are made on the median/mean value of
dimension with highest variance– each node stores one data point, leaves can be
empty
Example kD-tree
Use a form of A* search using the minimum distance to a node as an underestimate of the true closest distance
Finds closest neighbor in logarithmic (depth of tree) time
kD-trees (cont.)
Building a good kD-tree may take some time
– Learning time is no longer 0– Incremental learning is no longer trivial• kD-tree will no longer be balanced• re-building the tree is recommended when the max-
depth becomes larger than 2* the minimal required depth (= log(N) with N training examples)
Moving away from
lazy learning
Moving away from
lazy learning
Cover trees are more advanced, more complex, and more efficient!!
(4) Using Prototypes
The rough decision surfaces of nearest neighbor can sometimes be considered a disadvantage
– Solve two problems at once by using prototypes= Representative for a whole group of instances
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Prototypes (cont.)
Prototypes can be:– Single instance, replacing a group– Other structure (e.g., rectangle, rule, ...)
-> in this case: need to define distance
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Moving further and
further away from
lazy learning
Moving further and
further away from
lazy learning
Recommender Systems through instance based learning
Movie Alice (1) Bob (2) Carol (3) Dave (4)(romance) (action)
Love at last 5 5 0 0 0.9 0
Romance forever 5 ? ? 0 1.0 0.01
Cute puppies of love ? 4 0 ? 0.99 0
Nonstop car chases 0 0 5 4 0.1 1.0
Swords vs. karate 0 0 5 ? 0 0.9
Predict ratings for films users have not yet seen (or rated).
Recommender Systems
Predict through instance based regression:
Avg. rating
of user i can be allentries or kNN
rating by user jof entry k
Pearson coefficient
Some Comments on k-NN
Positive• Easy to implement• Good “baseline” algorithm /
experimental control• Incremental learning easy• Psychologically plausible
model of human memory
Negative• Led astray by irrelevant
features• No insight into domain (no
explicit model)• Choice of distance function
is problematic• Doesn’t exploit/notice
structure in examples
Summary
• Generalities of instance based learning– Basic idea, (dis)advantages, Voronoi diagrams,
lazy vs. eager learning
• Various instantiations– kNN, distance-weighted methods, ...– Rescaling attributes– Use of prototypes
Bayesian learning
This is going to be very introductory
•Describing (results of) learning processes– MAP and ML hypotheses
•Developing practical learning algorithms– Naïve Bayes learner• application: learning to classify texts
– Learning Bayesian belief networks
Bayesian approaches
Several roles for probability theory in machine learning:– describing existing learners• e.g. compare them with “optimal” probabilistic
learner
– developing practical learning algorithms• e.g. “Naïve Bayes” learner
Bayes’ theorem plays a central role
Basics of probability
• P(A): probability that A happens• P(A|B): probability that A happens, given that
B happens (“conditional probability”)• Some rules:– complement: P(not A) = 1 - P(A)– disjunction: P(A or B) = P(A)+P(B)-P(A and B)– conjunction: P(A and B) = P(A) P(B|A)
= P(A) P(B) if A and B independent– total probability:P(A) = i P(A|Bi) P(Bi)
With each Bi mutually exclusive
Bayes’ Theorem
P(A|B) = P(B|A) P(A) / P(B)
Mainly 2 ways of using Bayes’ theorem:– Applied to learning a hypothesis h from data D:
P(h|D) = P(D|h) P(h) / P(D) ~ P(D|h)P(h)– P(h): a priori probability that h is correct– P(h|D): a posteriori probability that h is correct– P(D): probability of obtaining data D– P(D|h): probability of obtaining data D if h is correct
– Applied to classification of a single example e:P(class|e) = P(e|class)P(class)/P(e)
Bayes’ theorem: Example
Example:– assume some lab test for a disease has 98%
chance of giving positive result if disease is present, and 97% chance of giving negative result if disease is absent
– assume furthermore 0.8% of population has this disease
– given a positive result, what is the probability that the disease is present?
P(Dis|Pos) = P(Pos|Dis)P(Dis) / P(Pos) = 0.98*0.008 / (0.98*0.008 + 0.03*0.992)
MAP and ML hypotheses
Task: Given the current data D and some hypothesis space H, return the hypothesis h in H that is most likely to be correct.
Note: this h is optimal in a certain sense– no method can exist that finds with higher
probability the correct h
MAP hypothesis
Given some data D and a hypothesis space H, find the hypothesis hH that has the highest probability of being correct; i.e., P(h|D) is maximal
This hypothesis is called the maximal a posteriori hypothesis hMAP : hMAP = argmaxhH P(h|D)
= argmaxhH P(D|h)P(h)/P(D) = argmaxhH P(D|h)P(h)
•last equality holds because P(D) is constant
So : we need P(D|h) and P(h) for all hH to compute hMAP
ML hypothesis
P(h): a priori probability that h is correct
What if no preferences for one h over another?•Then assume P(h) = P(h’) for all h, h’H •Under this assumption hMAP is called the maximum likelihood hypothesis hML
hML = argmaxhH P(D|h) (because P(h) constant)•How to find hMAP or hML ?– brute force method: compute P(D|h), P(h) for all hH– usually not feasible
Naïve Bayes classifier
Simple & popular classification method•Based on Bayes’ rule + assumption of conditional independence– assumption often violated in practice– even then, it usually works well
Example application: classification of text documents
Classification using Bayes rule
Given attribute values, what is most probable value of target variable?
Problem: too much data needed to estimate P(a1…an|vj)
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The Naïve Bayes classifier
Naïve Bayes assumption: attributes are independent, given the class
P(a1,…,an|vj) = P(a1|vj)P(a2|vj)…P(an|vj)–also called conditional independence (given the class)
•Under that assumption, vMAP becomes
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Learning a Naïve Bayes classifier
To learn such a classifier: just estimate P(vj), P(ai|vj) from data
How to estimate?– simplest: standard estimate from statistics• estimate probability from sample proportion• e.g., estimate P(A|B) as count(A and B) / count(B)
– in practice, something more complicated needed…
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Estimating probabilities
Problem:– What if attribute value ai never observed for class vj?
– Estimate P(ai|vj)=0 because count(ai and vj) = 0 ?• Effect is too strong: this 0 makes the whole product 0!
Solution: use m-estimate– interpolates between observed value nc/n and a priori
estimate p -> estimate may get close to 0 but never 0• m is weight given to a priori estimate
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Learning to classify text
Example application:– given text of newsgroup article, guess which
newsgroup it is taken from– Naïve bayes turns out to work well on this
application– How to apply NB?– Key issue : how do we represent examples? what
are the attributes?
Representation
Binary classification (+/-) or multiple classes possible
Attributes = word frequencies– Vocabulary = all words that occur in learning task– # attributes = size of vocabulary– Attribute value = word count or frequency in the
text (using m-estimate)= “Bag of Words” representation
Algorithm
procedure learn_naïve_bayes_text(E: set of articles, V: set of classes)Voc = all words and tokens occurring in Eestimate P(vj) and P(wk|vj) for all wk in E and vj in V:
Nj = number of articles of class jN = number of articlesP(vj) = Nj/Nnkj = number of times word wk occurs in text of class jnj = number of words in class j (counting doubles)P(wk|vj) = (nkj+1)/(nj+|Voc|)
procedure classify_naïve_bayes_text(A: article)remove from A all words/tokens that are not in Vocreturn argmaxvjV P(vj) i P(ai|vj)
procedure learn_naïve_bayes_text(E: set of articles, V: set of classes)Voc = all words and tokens occurring in Eestimate P(vj) and P(wk|vj) for all wk in E and vj in V:
Nj = number of articles of class jN = number of articlesP(vj) = Nj/Nnkj = number of times word wk occurs in text of class jnj = number of words in class j (counting doubles)P(wk|vj) = (nkj+1)/(nj+|Voc|)
procedure classify_naïve_bayes_text(A: article)remove from A all words/tokens that are not in Vocreturn argmaxvjV P(vj) i P(ai|vj)
Some (old) experimental results:– 1000 articles taken from 20 newsgroups– guess correct newsgroup for unseen documents– 89% classification accuracy with previous
approach
•Note: more recent approaches based on SVMs, … have been reported to work better– But Naïve Bayes still used in practice, e.g., for
spam detection
Bayesian Belief Networks
Consider two extremes of spectrum:– guessing joint probability distribution• would yield optimal classifier• but infeasible in practice (too much data needed)
– Naïve Bayes• much more feasible• but strong assumptions of conditional independence
•Is there something in between?– make some independence assumptions, but only
where reasonable
Bayesian belief networks
Bayesian belief network consists of1: graph• intuitively: indicates which variables “directly
influence” which other variables– arrow from A to B: A has direct effect on B– parents(X) = set of all nodes directly influencing X
• formally: each node is conditionally independent of each of its non-descendants, given its parents– conditional independence: cf. Naïve Bayes– X conditionally independent of Y given Z iff P(X|Y,Z) = P(X|Z)
2: conditional probability tables• for each node X : P(X|parents(X)) is given
Example
• Burglary or earthquake may cause alarm to go off• Alarm going off may cause one of neighbours to
call
Burglary Earthquake
Alarm
John calls Mary calls
B,E B,-E -B,E -B,-E A 0.9 0.8 0.4 0.01 -A 0.1 0.2 0.6 0.99
E 0.01-E 0.99
A -A M 0.9 0.2 -M 0.1 0.8
B 0.05-B 0.95
A -A J 0.8 0.1 -J 0.2 0.9
Network topology usually reflects direct causal influences–other structure also possible–but may render network more complex
Mary calls
Earthquake
Burglary
John calls
Alarm
Burglary Earthquake
Alarm
John calls Mary calls
Graph + conditional probability tables allow to construct joint probability distribution of all variables– P(X1,X2,…,Xn) = i P(Xi|parents(Xi))
– In other words: bayesian belief network carries full information on joint probability distribution
Inference
Given values for certain nodes, infer probability distribution for values of other nodes
•General algorithm quite complicated– See, e.g., Russel & Norvig, 1995: Artificial
Intelligence, a Modern Approach
General case
In general: inference is NP-complete– approximating methods, e.g. Monte-Carlo
to be predicted
evidence (observed)
unobserved
Learning bayesian networks
• Assume structure of network given:– only conditional probability tables to be learnt– training examples may include values for all
variables, or just for some of them– when all variables observable:• estimating probabilities as easy as for Naïve Bayes• e.g. estimate P(A|B,C) as count(A,B,C)/count(B,C)
– when not all variables observable:• methods based on gradient descent or EM
• When structure of network not given:– search for structure + tables• e.g. propose structure, learn tables• propose change to structure, relearn, see whether
better results
– active research topic
To remember
• Importance of Bayes’ theorem• MAP, ML, MDL– definitions, characterising learners from this
perspective, relationship MDL-MAP• Bayes optimal classifier, Gibbs classifier• Naïve Bayes: how it works, assumptions
made, application to text classification• Bayesian networks: representation, inference,
learning
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