Post on 02-Dec-2014
description
Integrating Bayesian Networks and Simpson’s Paradox in Data Mining
Alex FreitasUniversity of Kent
Ken McGarryUniversity of Sunderland
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Outline of the Talk
Introduction to Knowledge Discovery & Data Mining
Constructing Bayesian networks from data Simpson’s paradox Proposed method for integrating Bayesian
networks and Simpson’s paradox Conclusions
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Introduction
Data Mining consists of extracting patterns from data, and it is the core step of a knowledge discovery process
pre-proc data mining post-proc
Data interesting 22, M, 30K patterns 26, F, 55K IF (salary = high)
………. THEN (credit = good)
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The Knowledge Discovery Process– a popular definition
“Knowledge Discovery in Databases is the non-trivial process of identifying valid, novel, potentially useful, and ultimately understandable patterns in data”
(Fayyad et al. 1996) Focus on the quality of discovered patterns
– independent of the data mining algorithm
This definition is often quoted, but not very seriously taken into account
– A lot of research on discovering valid, accurate patterns
– Little research on discovering potentially useful patterns
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Criteria to Evaluate the “Interestingness” of Discovered Patterns
useful
novel, surprising
comprehensible
valid (accurate)
Amount of Research
Difficulty of measurement
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On the difficulty of discovering surprising patterns in data mining
Focus on maximizing accuracy leads to very accurate but useless rules, e.g. (Brin et al. 1997) – census data:– IF (person is pregnant) THEN (gender is female)– IF (age 5) THEN (employed = no)
(Tsumoto 2000) extracted 29,050 rules from a medical dataset. Out of these, just 220 (less than 1%) were considered interesting or surprising to the user
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Bayesian network example
A B
C
D
A Bayesian network represents potentially causal patterns, which tend to be more useful for intelligent decision making
Motivation for Integrating Bayesian Networks and Simpson’s Paradox
However, algorithms for constructing Bayesian networks from data were not designed to discover surprising patterns
Simpson’s paradox is surprising by nature
Causality + Surprisingness tends to improve Usefulness
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Constructing Bayesian Networks from Data
Methods based on conditional independence tests– Not scalable to datasets with many variables (attributes)
Methods based on search guided by a scoring function– Iteratively create candidate solutions (Bayesian networks) and
evaluate the quality of each created network using a scoring function, until a stopping criteria is satisfied
– Sequential methods consider a single candidate solution at a time
– Population-based methods consider many candidate solutions at a time
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Examples of sequential method– B algorithm starts with an empty network and at each iteration
adds, to the current candidate solution, the edge that maximizes the value of the scoring function
– K2 algorithm requires that the variables be ordered and the user specifies a parameter: the maximum number of parents of each variable in the network to be constructed
Both are greedy methods (local search), which offer no guarantee of finding the optimal network
Population-based methods are global search methods, but are stochastic, so again no guarantees
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Limitations of methods for constructing Bayesian networks from data (1)
Theoretical limitation (best possible algorithm & data)
Bayesian networks are Independence maps (I-maps) of the true probability distribution
– Every independence between variables represented in the network is an actual independence in the true probability distribution
– Dependences between variables represented in the network are not guaranteed to be actual dependences in the true probability distribution
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Limitations of methods for constructing Bayesian networks from data (2)
Practical limitations
The problem of constructing the optimal net is too complex in large datasets, so we have to use methods which do not guarantee the discovery of the optimal net
Sampling variation and/or noisy data may mislead the Bayesian network construction method, further contributing to the discovery of a sub-optimal net
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Simpson’s Paradox (Pearl 2000)
Overall E (recovered) E (not recov.) Total Recov. RateDrug (C) 20 20 40 50% No Drug (C) 16 24 40 40%Total 36 44 80
Males E (recovered) E (not recov.) Total Recov. RateDrug (C) 18 12 30 60%No Drug (C) 7 3 10 70%Total 25 15 40
Females E (recovered) E (not recov.) Total Recov. RateDrug (C) 2 8 10 20%No Drug (C) 9 21 30 30%Total 11 29 40
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Simpson’s Paradox as a Surprising Pattern
Event C (“cause”) increases the probability of event E (“effect”) in a given population but, at the same time, decreases the probability of E in every subpopulation
No paradox in terms of probability theory, it looks a “paradox” under a causal interpretation
– Gender is a confounder variable in the previous example
Although Simpson’s paradox is known by statisticians, occurrences of the paradox are surprising to users
There are algorithms that systematically find instances of the paradox in data and rank them in decreasing order of surprisingness (Fabris & Freitas 2006)
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The proposed method for integrating Bayesian networks and Simpson’s paradox
Basic Idea:– In a Bayesian network, the dependence denoted by
edge C E can be spurious, i.e., due to a confounding variable F
(for the previously discussed reasons)
Two approaches exploring this basic idea
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First Approach: paradox detection before network construction
First, run an algorithm that detects occurrences of Simpson’s paradox in data (Fabris & Freitas 2006)
– Produces a paradox list PL Modify Bayesian network construction algorithms to
take into account this list, biasing the algorithms against including network edges involving the paradox
Consider a potential dependence represented by the edge C E, where C is apparent cause of effect E– If variables C, E are associated in an occurrence of
Simpson’s paradox in PL, the algorithm is biased against including edge C E in the network
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Consider a greedy algorithm that starts with an empty network and adds one edge to the network at a time, guided by a scoring function
FOR EACH candidate edge A B
compute the score of the network if A B is added to the network
penalize score if there is an occurrence of the paradox in list PL involving pair of variables A, B
SELECT edge with highest score and add it to the network
proposed extension
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The same basic kind of extension can be applied to an Estimation of Distribution Algorithm – EDA is a population-based evolutionary algorithm– It evaluates a complete candidate solution (network) at once
FOR EACH candidate solution in the population
compute the score of the network represented by the candidate solution
penalize score in proportion to the number of paradox occurrences in list PL that are associated with direct dependences A B in the network
proposed extension
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Second Approach: paradox detection after network construction
First, construct a Bayesian network from data Use the network to “prune” the search space for the
Simpson’s paradox detection algorithm The algorithm will focus its search on the pairs of
variables for which there is a direct dependence (i.e., an edge A B ) in the Bayesian network
For each pair of such variables, the algorithm will try to find a third variable that acts as a confounder between those two variables
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Bayesian net variables considered by Simpson’s paradox detection algorithm, considering the Bayesian net
Cause Effect Is there a counfounder?
A C ? B C ? C D ?
A B
C
D
A paradox occurrence involving the above pairs of cause and effect variables would be even more surprising to the user, due to the structure of the network
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Limitation of the proposed integration method
It is possible that the data does not contain any occurrence of Simpson’s paradox– In this case the usefulness of the method is limited
Even if the algorithm does not find any paradox occurrence, this result is to some extent useful:
– it gives us increased confidence that the dependences represented in the network are true dependences, rather than spurious ones
– This additional test complements (rather than replaces) conventional methods for evaluating Bayesian networks
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Conclusions
We proposed a method for integrating two very different kinds of algorithm in data mining
– Algorithms for constructing Bayesian networks Discover potentially causal, more useful patterns
– Algorithms for detecting Simpson’s paradox Discover surprising patterns, potentially more useful
Hopefully, combining the “best of both worlds”, increasing the chance of discovering patterns useful for intelligent decision making by the user
Future research: computational implementation of the proposed method and analysis of results
Any Questions ??
Thanks for listening!