Alleviating Naive Bayes Attribute Independence Assumption

42 Pages·2013·0.42 MB·English

JournalofMachineLearningResearch14(2013)1947-1988 Submitted6/12;Revised1/13;Published7/13
Alleviating Naive Bayes Attribute Independence Assumption by
Attribute Weighting
NayyarA.Zaidi [email protected]
FacultyofInformationTechnology
MonashUniversity
VIC3800,Australia
Jesu´sCerquides [email protected]
IIIA-CSIC,ArtificialIntelligenceResearchInstitute
SpanishNationalResearchCouncil
CampusUAB
08193Bellaterra,Spain
MarkJ.Carman [email protected]
GeoffreyI.Webb [email protected]
FacultyofInformationTechnology
MonashUniversity
VIC3800,Australia
Editor:RussGreiner
Abstract
DespitethesimplicityoftheNaiveBayesclassifier,ithascontinuedtoperformwellagainstmore
sophisticated newcomers and has remained, therefore, of great interest to the machine learning
community. OfnumerousapproachestorefiningthenaiveBayesclassifier,attributeweightinghas
receivedlessattentionthanitwarrants.Mostapproaches,perhapsinfluencedbyattributeweighting
inothermachinelearningalgorithms, useweightingtoplacemoreemphasisonhighlypredictive
attributesthanthosethatarelesspredictive. Inthispaper,wearguethatfornaiveBayesattribute
weightingshouldinsteadbeusedtoalleviatetheconditionalindependenceassumption. Basedon
thispremise,weproposeaweightednaiveBayesalgorithm,calledWANBIA,thatselectsweights
tominimizeeitherthenegativeconditionalloglikelihoodorthemeansquarederrorobjectivefunc-
tions. WeperformextensiveevaluationsandfindthatWANBIAisacompetitivealternativetostate
oftheartclassifierslikeRandomForest,LogisticRegressionandA1DE.
Keywords: classification,naiveBayes,attributeindependenceassumption,weightednaiveBayes
classification
1. Introduction
NaiveBayes(alsoknownassimpleBayesandIdiot’sBayes)isanextremelysimpleandremarkably
effectiveapproachtoclassificationlearning(Lewis,1998;HandandYu,2001). Itinferstheproba-
bility of a class label given data using a simplifying assumption that the attributes are independent
giventhelabel(Kononenko,1990;Langleyetal.,1992). Thisassumptionismotivatedbytheneed
toestimatehigh-dimensionalmulti-variateprobabilitiesfromthetrainingdata. Ifthereissufficient
data present for every possible combination of attribute values, direct estimation of each relevant
multi-variate probability will be reliable. In practice, however, this is not the case and most com-
c2013NayyarA.Zaidi,Jesu´sCerquides,MarkJ.CarmanandGeoffreyI.Webb.
(cid:13)
ZAIDI,CERQUIDES,CARMANANDWEBB
binations are either not represented in the training data or not present in sufficient numbers. Naive
Bayes circumvents this predicament by its conditional independence assumption. Surprisingly, it
has been shown that the prediction accuracy of naive Bayes compares very well with other more
complex classifiers such as decision trees, instance-based learning and rule learning, especially
when the data quantity is small (Hand and Yu, 2001; Cestnik et al., 1987; Domingos and Pazzani,
1996;Langleyetal.,1992).
In practice, naive Bayes’ attribute independence assumption is often violated, and as a result
its probability estimates are often suboptimal. A large literature addresses approaches to reducing
theinaccuraciesthatresultfromtheconditionalindependenceassumption. Suchapproachescanbe
placedintotwocategories. Thefirstcategorycomprisessemi-naiveBayesmethods. Thesemethods
are aimed at enhancing naive Bayes’ accuracy by relaxing the assumption of conditional indepen-
dencebetweenattributesgiventheclasslabel(LangleyandSage,1994;FriedmanandGoldszmidt,
1996;Zhengetal.,1999;CerquidesandDeMa´ntaras,2005a;Webbetal.,2005,2011;Zhengetal.,
2012). Thesecondcategorycomprisesattributeweightingmethodsandhasreceivedrelativelylittle
attention(HildenandBjerregaard,1976;Ferreiraetal.,2001;Hall,2007). Thereissomeevidence
that attribute weighting appears to have primarily been viewed as a means of increasing the influ-
enceofhighlypredictiveattributesanddiscountingattributesthathavelittlepredictivevalue. This
isnotsomuchevidentfromtheexplicitmotivationstatedinthepriorwork,butratherfromtheman-
ner in which weights have been assigned. For example, weighting by mutual information between
an attribute and the class is directly using a measure of how predictive is each individual attribute
(Zhang and Sheng, 2004). In contrast, we argue that the primary value of attribute weighting is its
capacitytoreducetheimpactonpredictionaccuracyofviolationsoftheassumptionofconditional
attributeindependence.
Contributionsofthispaperaretwo-fold:
ThispaperreviewsthestateoftheartinweightednaiveBayesianclassification. Weprovidea
•
compactsurveyofexistingtechniquesandcomparethemusingthebias-variancedecomposi-
tionmethodofKohaviandWolpert(1996). WealsouseFriedmantestandNemenyistatistics
toanalyzeerror,bias,varianceandrootmeansquareerror.
WepresentnovelalgorithmsforlearningattributeweightsfornaiveBayes. Itshouldbenoted
•
that the motivation of our work differs from most previous attribute weighting methods. We
viewweightingasawaytoreducetheeffectsoftheviolationsoftheattributeindependence
assumption on which naive Bayes is based. Also, our work differs from semi-naive Bayes
methods,asweweighttheattributesratherthanmodifyingthestructureofnaiveBayes.
WeproposeaweightednaiveBayesalgorithm,WeightingattributestoAlleviateNaiveBayes’Inde-
pendenceAssumption(WANBIA),thatintroducesweightsinnaiveBayesandlearnstheseweights
in a discriminative fashion that is minimizing either the negative conditional log likelihood or the
mean squared error objective functions. Naive Bayes probabilities are set to be their maximum a
posteriori(MAP)estimates.
Thepaperisorganizedasfollows: weprovideaformaldescriptionoftheweightednaiveBayes
model in Section 2. Section 3 provides a survey of related approaches. Our novel techniques for
learning naive Bayes weights are described in Section 4 where we also discuss their connection
with naive Bayes and Logistic Regression in terms of parameter optimization. Section 5 presents
experimental evaluation of our proposed methods and their comparison with related approaches.
Section6presentsconclusionsanddirectionsforfutureresearch.
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ALLEVIATINGNBATTRIBUTEINDEPENDENCEASSUMPTIONBYATTRIBUTEWEIGHTING
Notation Description
P(e) theunconditionedprobabilityofevente
P(e g) conditionalprobabilityofeventegiveng
|
Pˆ( ) anestimateofP( )
• •
a thenumberofattributes
n thenumberofdatapointsinD
x= x ,...,x anobject(a-dimensionalvector)andx D
1 a
h i ∈
y Y theclasslabelforobjectx
∈
Y thenumberofclasses
| |
D = x(1),...,x(n) dataconsistingofnobjects
{ }
L = y(1),...,y(n) labelsofdatapointsinD
{ }
X discretesetofvaluesforattributei
i
X thecardinalityofattributei
i
| |
v= 1∑ X theaveragecardinalityoftheattributes
a i| i|
Table1: Listofsymbolsused
2. WeightedNaiveBayes
We wish to estimate from a training sample D consisting of n objects, the probability P(y x) that
|
anexamplex D belongstoaclasswithlabely Y. Allthesymbolsusedinthisworkarelisted
∈ ∈
inTable1. Fromthedefinitionofconditionalprobabilitywehave
P(y x)=P(y,x)/P(x). (1)
|
Y
AsP(x)=∑i|=1|P(yi,x),wecanalwaysestimateP(y|x)inEquation1fromtheestimatesofP(y,x)
foreachclassas:
P(y,x)
P(y,x)/P(x)= . (2)
Y
∑i|=1|P(yi,x)
Inconsequence,intheremainderofthispaperweconsideronlytheproblemofestimatingP(y,x).
NaiveBayesestimatesP(y,x)byassumingtheattributesareindependentgiventheclass,result-
inginthefollowingformula:
a
Pˆ(y,x)=Pˆ(y)∏Pˆ(x y). (3)
i
|
i=1
Weighted naive Bayes extends the above by adding a weight to each attribute. In the most general
case,thisweightdependsontheattributevalue:
a
Pˆ(y,x)=Pˆ(y)∏Pˆ(xi y)wi,xi. (4)
|
i=1
Doing this results in ∑a X weight parameters (and is in some cases equivalent to a “binarized
i i
| |
logistic regression model” see Section 4 for a discussion). A second possibility is to give a single
weightperattribute:
a
Pˆ(y,x)=Pˆ(y)∏Pˆ(x y)wi. (5)
i
|
i=1
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Onefinalpossibilityistosetallweightstoasinglevalue:
w
a
Pˆ(y,x)=Pˆ(y) ∏Pˆ(x y) . (6)
i
|
i=1 !
Equation 5 is a special case of Equation 4, where w =w, and Equation 6 is a special case
i,j ij i
∀
of Equation 5 where w =w. Unless explicitly stated, in this paper we intend the intermediate
i i
∀
formwhenwerefertoattributeweighting, aswebelieveitprovidesaneffectivetrade-offbetween
computationalcomplexityandinductivepower.
AppropriateweightscanreducetheerrorthatresultsfromviolationsofnaiveBayes’conditional
attribute independence assumption. Trivially, if data include a set of a attributes that are identical
to one another, the error due to the violation of the conditional independence assumption can be
removed by assigning weights that sum to 1.0 to the set of attributes in the set. For example, the
weight for one of the attributes, x could be set to 1.0, and that of the remaining attributes that are
i
identical to x set to 0.0. This is equivalent to deleting the remaining attributes. Note that, any
i
assignment of weights such that their sum is 1.0 for the a attributes will have the same effect, for
example,wecouldsettheweightsofallaattributesto1/a.
Attribute weighting is strictly more powerful than attribute selection, as it is possible to obtain
identical results to attribute selection by setting the weights of selected attributes to 1.0 and of
discarded attributes to 0.0, and assignment of other weights can create classifiers that cannot be
expressedusingattributeselection.
2.1 DealingwithDependentAttributesbyWeighting: ASimpleExample
ThisexampleshowstherelativeperformanceofnaiveBayesandweightednaiveBayesaswevary
the conditional dependence between attributes. In particular it demonstrates how optimal assign-
ment of weights will never result in higher error than attribute selection or standard naive Bayes,
and that for certain violations of the attribute independence assumption it can result in lower error
thaneither.
We will constrain ourselves to a binary class problem with two binary attributes. We quantify
theconditionaldependencebetweentheattributesusingtheConditionalMutualInformation(CMI):
P(x ,x y)
I(X ,X Y) = ∑∑∑P(x ,x ,y)log 1 2| .
1 2 1 2
| P(x y)P(x y)
y x2 x1 1| 2|
Theresultsofvaryingtheconditionaldependencebetweentheattributesontheperformanceofthe
differentclassifiersintermsoftheirRootMeanSquaredError(RMSE)isshowninFigure1.
Togeneratethesecurves,wevariedtheprobabilitiesP(y x ,x )andP(x ,x )andplottedaverage
1 2 1 2
|
results across distinct values of the Conditional Mutual Information. For each of the 4 possible
attributevaluecombinations(x ,x ) (0,0),(0,1),(1,0),(1,1) ,weselectedvaluesfortheclass
1 2
∈{ }
probability given the attribute value combination from the set: P(y x ,x ) 0.25,0.75 . Note
1 2
| ∈ { }
that P( y x ,x )=1 P(y x ,x ), so this process resulted in 24 possible assignments to the vector
1 2 1 2
¬ | − |
P(y , ).
|• •
We then set the values for the attribute value probabilities P(x ,x ) by fixing the marginal dis-
1 2
tributions to a half P(x )=P(x )=1/2, and varying the correlation between the attributes using
1 2
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ALLEVIATINGNBATTRIBUTEINDEPENDENCEASSUMPTIONBYATTRIBUTEWEIGHTING
0.05
Prior
Naive Bayes
0.045 Selective Naive Bayes
Weighted Naive Bayes
0.04
0.035
0.03
Error0.025
0.02
0.015
0.01
0.005
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Conditional Dependence
Figure1: VariationofErrorofnaiveBayes,selectivenaiveBayes,weightednaiveBayesandclassi-
fierbasedonlyonpriorprobabilitiesoftheclassasafunctionofconditionaldependence
(conditionalmutualinformation)betweenthetwoattributes.
Pearson’scorrelationcoefficient,denotedρ,asfollows:1
(1+ρ)
P(X =0,X =0) = P(X =1,X =1)= ,
1 2 1 2
4
(1 ρ)
P(X =0,X =1) = P(X =1,X =0)= − ,
1 2 1 2
4
where 1 ρ 1.
− ≤ ≤
Notethatwhenρ= 1theattributesareperfectlyanti-correlated(x = x ),whenρ=0theattributes
1 2
− ¬
are independent (since the joint distribution P(x ,x ) is uniform) and when ρ=1 the attributes are
1 2
perfectlycorrelated.
For the graph, we increased values of ρ in increments of 0.00004, resulting in 50000 distribu-
tions(vectors)forP( , )foreachvectorP(y , ). Nearoptimalweights(w ,w )fortheweighted
1 2
• • |• •
naive Bayes classifier were found using grid search over the range 0.0,0.1,0.2,...,0.9,1.0
{{ }×
0.0,0.1,0.2,...,0.9,1.0 . Results in Figure 1 are plotted by taking average across conditional
{ }}
mutualinformationvalues,withawindowsizeof0.1.
1.NotethatfromthedefinitionofPearson’scorrelationcoefficientwehave:
ρ= E[(X1−E[X1])(X2−E[X2])] =4P(X1=1,X2=1) 1,
E[(X1 E[X1])2]E[(X2 E[X2])2] −
− −
sinceE[X1]=E[X2]=P(1)p=1/2andE[X1X2]=P(X1=1,X2=1).
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ZAIDI,CERQUIDES,CARMANANDWEBB
WecomparetheexpectedRMSEofnaiveBayes(w =1,w =1), weightednaiveBayes, naive
1 2
Bayes based on feature 1 only (selective Bayes with w =1,w =0), naive Bayes based on feature
1 2
2 only (selective Bayes with w =0,w =1), and naive Bayes using only the prior (equivalent to
1 2
weighted naive Bayes with both weights set to 0.0). It can be seen that when conditional mutual
information (CMI) is small, naive Bayes performs better than selective naive Bayes and the prior
classifier. Indeed, when CMI is 0.0, naive Bayes is optimal. As CMI is increased, naive Bayes
performance deteriorates compared to selective naive Bayes. Weighted naive Bayes, on the other
hand, hasthebestperformanceinallcircumstances. Duetothesymmetryoftheproblem, thetwo
selectiveBayesclassifiersgiveexactlythesameresults.
Notethatinthisexperimentwehaveusedtheoptimalweightstocalculatetheresults. Wehave
shown that weighted naive Bayes is capable of expressing more accurate classifiers than selective
naive Bayes. In theremaining sections we will examine and evaluatetechniques for learning from
datatheweightsthosemodelsrequire.
3. SurveyofAttributeWeightingandSelectingMethodsforNaiveBayes
Attribute weighting is well-understood in the context of nearest-neighbor learning methods and is
usedforreducingbiasinhigh-dimensionalproblemsduetothepresenceofredundantorirrelevant
features(Friedman,1994;Guyonetal.,2004). Itisalsousedformitigatingtheeffectsofthecurse-
of-dimensionalitywhichresultsinexponentialincreaseintherequiredtrainingdataasthenumber
offeaturesareincreased(Bellman,1957). AttributeweightingfornaiveBayesiscomparativelyless
explored.
Before discussing these techniques, however, it is useful to briefly examine the closely related
area of feature selection for naive Bayes. As already pointed out, weighting can achieve feature
selectionbysettingsweightstoeither0.0or1.0,andsocanbeviewedasageneralizationoffeature
selection.
Langley and Sage (1994) proposed the Selective Bayes (SB) classifier, using feature selection
to accommodate redundant attributes in the prediction process and to augment naive Bayes with
the ability to exclude attributes that introduce dependencies. The technique is based on searching
through the entire space of all attribute subsets. For that, they use a forward sequential search
with a greedy approach to traversethe search space. That is, the algorithm initializes the subsetof
attributestoanemptyset,andtheaccuracyoftheresultingclassifier,whichsimplypredictsthemost
frequentclass,issavedforsubsequentcomparison. Oneachiteration,themethodconsidersadding
each unused attribute to the subset on a trial basis and measures the performance of the resulting
classifier on the training data. The attribute that most improves the accuracy is permanently added
tothesubset. Thealgorithmterminateswhenadditionofanyattributeresultsinreducedaccuracy,
atwhichpointitreturnsthelistofcurrentattributesalongwiththeirranks. Therankoftheattribute
isbasedontheorderinwhichtheyareaddedtothesubset.
Similar to Langley and Sage (1994), Correlation-based Feature Selection (CFS) used a corre-
lation measure as a metric to determine the relevance of the attribute subset (Hall, 2000). It uses
a best-first search to traverse through feature subset space. Like SB, it starts with an empty set
andgeneratesallpossiblesinglefeatureexpansions. Thesubsetwithhighestevaluationisselected
and expanded in the same manner by adding single features. If expanding a subset results in no
improvement, the search drops back to the next best unexpanded subset and continues from there.
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ALLEVIATINGNBATTRIBUTEINDEPENDENCEASSUMPTIONBYATTRIBUTEWEIGHTING
Thebestsubsetfoundisreturnedwhenthesearchterminates. CFSusesastoppingcriterionoffive
consecutivefullyexpandednon-improvingsubsets.
Therehasbeenagrowingtrendintheuseofdecisiontreestoimprovetheperformanceofother
learningalgorithmsandnaiveBayesclassifiersarenoexception. Forexample,onecanbuildanaive
BayesclassifierbyusingonlythoseattributesappearinginaC4.5decisiontree. Thisisequivalentto
givingzeroweightstoattributesnotappearinginthedecisiontree. TheSelectiveBayesianClassifier
(SBC)ofRatanamahatanaandGunopulos(2003)alsoemploysdecisiontreesforattributeselection
fornaiveBayes. Onlythoseattributesappearinginthetopthreelevelsofadecisiontreeareselected
forinclusioninnaiveBayes. Sincedecisiontreesareinherentlyunstable,fivedecisiontrees(C4.5)
are generated on samples generated by bootstrapping 10% from the training data. Naive Bayes is
trained on an attribute set which comprises the union of attributes appearing in all five decision
trees.
One of the earliest works on weighted naive Bayes is by Hilden and Bjerregaard (1976), who
used weighting of the form of Equation 6. This strategy uses a single weight and therefore is not
strictly performing attribute weighting. Their approach is motivated as a means of alleviating the
effectsofviolationsoftheattributeindependenceassumption. Settingwtounityisappropriatewhen
the conditional independence assumption is satisfied. However, on their data set (acute abdominal
painstudyinCopenhagenbyBjerregaardetal.1976),improvedclassificationwasobtainedwhenw
wassmall,withanoptimumvalueaslowas0.3. Theauthorspointoutthatifsymptomvariablesof
aclinicalfieldtrialarenotindependent,butpair-wisecorrelatedwithindependencebetweenpairs,
then w=0.5 will be the correct choice since using w=1 would make all probabilities the square
of what they ought be. Looking at the optimal value of w=0.3 for their data set, they suggested
that out of ten symptoms, only three are providing independent information. The value of w was
obtainedbymaximizingthelog-likelihoodovertheentiretestingsample.
Zhang and Sheng (2004) used the gain ratio of an attribute with the class labels as its weight.
Their formula is shown in Equation 7. The gain ratio is a well-studied attribute weighting tech-
nique and is generally used for splitting nodes in decision trees (Duda et al., 2006). The weight
of each attribute is set to the gain ratio of the attribute relative to the average gain ratio across all
attributes. Notethat, asaresultofthedefinitionatleastone(possiblymany)oftheattributeshave
weights greater than 1, which means that they are not only attempting to lessen the effects of the
independenceassumption—otherwisetheywouldrestricttheweightstobenomorethanone.
GR(i)
w = . (7)
i 1∑a GR(i)
a i=1
The gain ratio of an attribute is then simply the Mutual Information between that attribute and the
classlabeldividedbytheentropyofthatattribute:
GR(i)= I(Xi,Y) = ∑y∑x1P(x1,y)logPP(x(1x)1P,y()y).
H(Xi) ∑x1P(x1)logP(1x1)
Severalotherwrapper-basedmethodsarealsoproposedinZhangandSheng(2004). Forexample,
they use a simple hill climbing search to optimize weight w using Area Under Curve (AUC) as an
evaluationmetric. AnotherMarkov-Chain-Monte-Carlo(MCMC)methodisalsoproposed.
AnattributeweightingschemebasedondifferentialevolutionalgorithmsfornaiveBayesclas-
sificationhavebeenproposedinWuandCai(2011). First,apopulationofattributeweightvectors
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is randomly generated, weights in the population are constrained to be between 0 and 1. Second,
typicalgeneticalgorithmicstepsofmutationandcross-overareperformedoverthethepopulation.
They defined a fitness function which is used to determine if mutation can replace the current in-
dividual (weight vector) with a new one. Their algorithm employs a greedy search strategy, where
mutatedindividualsareselectedasoffspringonlyifthefitnessisbetterthanthatoftargetindividual.
Otherwise,thetargetismaintainedinthenextiteration.
AschemeusedinHall(2007)issimilarinspirittoSBCwheretheweightassignedtoeachat-
tributeisinverselyproportionaltotheminimumdepthatwhichtheywerefirsttestedinanunpruned
decisiontree. Weightsarestabilizedbyaveragingacross10decisiontreeslearnedondatasamples
generated by bootstrapping 50% from the training data. Attributes not appearing in the decision
treesareassignedaweightofzero. Forexample,onecanassignweighttoanattributeias:
1 T 1
w = ∑ . (8)
i
T √d
t ti
where d is the minimum depth at which the attribute i appears in decision tree t, and T is the
ti
total number of decision trees generated. To understand whether the improvement in naive Bayes
accuracy was due to attribute weighting or selection, a variant of the above approach was also
proposed where all non-zero weights are set to one. This is equivalent to SBC except using a
bootstrapsizeof50%with10iterations.
BothSBandCFSarefeatureselectionmethods. Sinceselectinganoptimalnumberoffeatures
is not trivial, Hall (2007) proposed to use SB and CFS for feature weighting in naive Bayes. For
example,theweightofanattributeicanbedefinedas:
1
w = . (9)
i
√ri
wherer istherankofthefeaturebasedonSBandCFSfeatureselection.
i
The feature weighting method proposed in Ferreira et al. (2001) is the only one to use Equa-
tion 4, weighting each attribute value rather than each attribute. They used entropy-based dis-
cretization for numeric attributes and assigned a weight to each partition (value) of the attribute
thatisproportionaltoitspredictivecapabilityoftheclass. Differentweightfunctionsareproposed
to assign weights to the values. These functions measure the difference between the distribution
overclassesfortheparticularattribute-valuepairanda“baselineclassdistribution”. Thechoiceof
weight function reduces to a choice of baseline distribution and the choice of measure quantifying
thedifferencebetweenthedistributions. Theyusedtwosimplebaselinedistributionschemes. The
firstassumesequiprobableclasses,thatis,uniformclasspriors. Inthatcasetheweightofforvalue
joftheattributeicanbewrittenas:
1/α
1
w ∝ ∑ P(y X =j) α .
ij | | i − Y |
y | | !
where P(y X = j) denotes the probability that the class is y given that the i-th attribute of a data
i
|
point has value j. Alternatively, the baseline class distribution can be set to the class probabilities
acrossallvaluesoftheattribute(i.e.,theclasspriors). Theweighingfunctionwilltaketheform:
1/α
w ∝ ∑ P(y X =j) P(y X =miss) α .
ij i i
| | − | 6 |
y !
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ALLEVIATINGNBATTRIBUTEINDEPENDENCEASSUMPTIONBYATTRIBUTEWEIGHTING
where P(y X =miss) is the class prior probability across all data points for which the attribute i
i
| 6
is not missing. Equation 10 and 10 assume an L distance metric where α=2 corresponds to the
α
L norm. Similarly, theyhavealsoproposedtousedistancebasedonKullback-Leiblerdivergence
2
betweenthetwodistributionstosetweights.
Many researchers have investigated techniques for extending the basic naive Bayes indepen-
dence model with a small number of additional dependencies between attributes in order to im-
prove classification performance (Zheng and Webb, 2000). Popular examples of such semi-naive
BayesmethodsincludeTree-AugmentedNaiveBayes(TAN)(Friedmanetal.,1997)andensemble
methods such as Averaged n-Dependence Estimators (AnDE) (Webb et al., 2011). While detailed
discussionofthesemethodsisbeyondthescopeofthiswork,wewilldescribebothTANandAnDE
inSection5.10forthepurposesofempiricalcomparison.
Semi-naive Bayes methods usually limit the structure of the dependency network to simple
structuressuchastrees,butmoregeneralgraphstructurescanalsobelearnt. Considerableresearch
hasbeendoneintheareaoflearninggeneralBayesianNetworks(Greineretal.,2004;Grossmanand
Domingos, 2004; Roos et al., 2005), with techniques differing on whether the network structure is
chosentooptimizeagenerativeordiscriminativeobjectivefunction,andwhetherthesameobjective
isalsousedforoptimizingtheparametersofthemodel. Indeedoptimizingnetworkstructureusing
adiscriminativeobjectivefunctioncanquicklybecomecomputationallychallengingandthusrecent
work in this area has looked at efficient heuristics for discriminative structure learning (Pernkopf
and Bilmes, 2010) and at developing decomposable discriminative objective functions (Carvalho
etal.,2011).
In this paper we are interested in improving performance of the NB classifier by reducing the
effectofattributeindependenceviolationsthroughattributeweighting. Wedonotattempttoidentify
theparticulardependenciesbetweenattributesthatcausetheviolationsandthusarenotattempting
to address the much harder problem of inducing the dependency network structure. While it is
conceivablethatsemi-naiveBayesmethodsandmoregeneralBayesianNetworkclassifierlearning
couldalsobenefitfromattributeweighting,weleaveitsinvestigationtofuturework.
AsummaryofdifferentmethodscomparedinthisresearchisgiveninTable2.
4. WeightingtoAlleviatetheNaiveBayesIndependenceAssumption
Inthissection,wewilldiscussourproposedmethodstoincorporateweightsinnaiveBayes.
4.1 WANBIA
Many previous approaches to attribute weighting for naive Bayes have found weights using some
form of mechanism that increases the weights of attributes that are highly predictive of the class
and decreases the weights of attributes that are less predictive of the class. We argue that this is
not appropriate. Naive Bayes delivers Bayes optimal classification if the attribute independence
assumptionholds. Weightingshouldonlybeappliedtoremedyviolationsoftheattributeindepen-
dence assumption. For example, consider the case where there are three attributes, x , x and x ,
1 2 3
such that x and x are conditionally independent of one another given the class and x is an exact
1 2 3
copy of x (and hence violates the independence assumption). Irrespective of any measure of how
1
well these three attributes each predict the class, Bayes optimal classification will be obtained by
setting the weights of x and x to sum to 1.0 and setting the weight of x to 1.0. In contrast, a
1 3 2
method that uses a measure such as mutual information with the class to weight the attribute will
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ZAIDI,CERQUIDES,CARMANANDWEBB
Name Description
NaiveBayes.
NB NaiveBayesClassifier.
WeightedNaiveBayes(usingTypicalFeatureWeightingMethods).
GRW UsegainratioasattributeweightsinnaiveBayes,showninEquation7(ZhangandSheng,2004).
SBC AssignweighttoattributeiasgiveninEquation8whereL=5withabootstrapsizeof10%. Alsodi=0if
di>3(RatanamahatanaandGunopulos,2003).
MH AssignweighttoattributeiasgiveninEquation8whereL=10withabootstrapsizeof50%(Hall,2007).
SB UseSelectiveBayesmethodtodeterminetherankofindividualfeaturesandassignweightsaccordingtoEqua-
tion9(LangleyandSage,1994).
CFS Usecorrelation-basedfeatureselectionmethodtodeterminetherankofindividualfeaturesandassignweights
accordingtoEquation9(LangleyandSage,1994;Hall,2007).
SelectiveNaiveBayes(usingTypicalFeatureSelectionMethods).
SBC-FS SimilartoSBCexceptwi=1ifwi>0.
MH-FS SimilartoMHexceptwi=1ifwi>0(Hall,2007).
WeightedNaiveBayes(Ferreiraetal.,2001).
FNB-d1 WeightscomputedperattributevalueusingEquation10withα=2.
FNB-d2 WeightscomputedperattributevalueusingEquation10withα=2.
Semi-naiveBayesClassifiers.
AnDE Averagen-DependentEstimator(Webbetal.,2011).
TAN TreeAugmentedNaiveBayes(Friedmanetal.,1997).
StateoftheArtClassificationTechniques.
RF RandomForests(Breiman,2001).
LR LogisticRegression(Roosetal.,2005).
WeightedNaiveBayes(ProposedMethods,willbediscussedinSection4).
CLL
WANBIA NaiveBayesweightsobtainedbymaximizingConditionalLog-Likelihood.
MSE
WANBIA NaiveBayesweightsobtainedbyminimizingMean-Square-Error.
Table2: Summaryoftechniquescomparedinthisresearch.
reducetheaccuracyoftheclassifierrelativetousinguniformweightsinanysituationwherex and
1
x receivehigherweightsthanx .
3 2
Ratherthanselectingweightsbasedonmeasuresofpredictiveness,wesuggestitismoreprof-
itable to pursue approaches such as those of Zhang and Sheng (2004) and Wu and Cai (2011) that
optimizetheweightstoimprovethepredictionperformanceoftheweightedclassifierasawhole.
FollowingfromEquations1,2and5,letusre-definetheweightednaiveBayesmodelas:
π ∏ θwi
Pˆ(y|x;π,Θ,w)= ∑ πy ∏i Xθi=wixi|y , (10)
y′ y′ i Xi=xi|y′
withconstraints:
∑ π =1 and ∑ θ =1,
y y ∀y,i j Xi=xi|y
where
π ,θ arenaiveBayesparameters.
• { y Xi=xi|y}
π [0,1] Y isaclassprobabilityvector.
| |
• ∈
ThematrixΘconsistofclassandattribute-dependentprobabilityvectorsθi,y [0,1]|Xi|.
• ∈
wisavectorofclass-independentweights,w foreachattributei.
i
•
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