Naive Bayes Classifier From Wikipedia

8/10/2019 Naive Bayes Classifier From Wikipedia

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Naive Bayes classifier

From Wikipedia, the free encyclopedia

Machine learninganddata mining

Problems

Classification

Clustering

Regression

Anomaly detection

Association rules

Reinforcement learning

Structured prediction

Feature learning

Online learning

Semi-supervised learning

Grammar induction

Supervised learning

(classificationregression)

Decision trees

Ensembles(Bagging,Boosting,Random forest)

k-NN

Linear regression

Naive Bayes

Neural networks

Logistic regression

Perceptron
http://en.wikipedia.org/wiki/Machine_learninghttp://en.wikipedia.org/wiki/Machine_learninghttp://en.wikipedia.org/wiki/Data_mininghttp://en.wikipedia.org/wiki/Data_mininghttp://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/wiki/Cluster_analysishttp://en.wikipedia.org/wiki/Cluster_analysishttp://en.wikipedia.org/wiki/Regression_analysishttp://en.wikipedia.org/wiki/Regression_analysishttp://en.wikipedia.org/wiki/Anomaly_detectionhttp://en.wikipedia.org/wiki/Anomaly_detectionhttp://en.wikipedia.org/wiki/Association_rule_learninghttp://en.wikipedia.org/wiki/Association_rule_learninghttp://en.wikipedia.org/wiki/Reinforcement_learninghttp://en.wikipedia.org/wiki/Reinforcement_learninghttp://en.wikipedia.org/wiki/Structured_predictionhttp://en.wikipedia.org/wiki/Structured_predictionhttp://en.wikipedia.org/wiki/Feature_learninghttp://en.wikipedia.org/wiki/Feature_learninghttp://en.wikipedia.org/wiki/Online_machine_learninghttp://en.wikipedia.org/wiki/Online_machine_learninghttp://en.wikipedia.org/wiki/Semi-supervised_learninghttp://en.wikipedia.org/wiki/Semi-supervised_learninghttp://en.wikipedia.org/wiki/Grammar_inductionhttp://en.wikipedia.org/wiki/Grammar_inductionhttp://en.wikipedia.org/wiki/Supervised_learninghttp://en.wikipedia.org/wiki/Supervised_learninghttp://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/wiki/Regression_analysishttp://en.wikipedia.org/wiki/Regression_analysishttp://en.wikipedia.org/wiki/Regression_analysishttp://en.wikipedia.org/wiki/Decision_tree_learninghttp://en.wikipedia.org/wiki/Decision_tree_learninghttp://en.wikipedia.org/wiki/Ensemble_learninghttp://en.wikipedia.org/wiki/Ensemble_learninghttp://en.wikipedia.org/wiki/Bootstrap_aggregatinghttp://en.wikipedia.org/wiki/Bootstrap_aggregatinghttp://en.wikipedia.org/wiki/Bootstrap_aggregatinghttp://en.wikipedia.org/wiki/Boosting_%28machine_learning%29http://en.wikipedia.org/wiki/Boosting_%28machine_learning%29http://en.wikipedia.org/wiki/Boosting_%28machine_learning%29http://en.wikipedia.org/wiki/Random_foresthttp://en.wikipedia.org/wiki/Random_foresthttp://en.wikipedia.org/wiki/Random_foresthttp://en.wikipedia.org/wiki/K-nearest_neighbors_classificationhttp://en.wikipedia.org/wiki/K-nearest_neighbors_classificationhttp://en.wikipedia.org/wiki/K-nearest_neighbors_classificationhttp://en.wikipedia.org/wiki/Linear_regressionhttp://en.wikipedia.org/wiki/Linear_regressionhttp://en.wikipedia.org/wiki/Artificial_neural_networkhttp://en.wikipedia.org/wiki/Artificial_neural_networkhttp://en.wikipedia.org/wiki/Logistic_regressionhttp://en.wikipedia.org/wiki/Logistic_regressionhttp://en.wikipedia.org/wiki/Perceptronhttp://en.wikipedia.org/wiki/Perceptronhttp://en.wikipedia.org/wiki/File:Linear-svm-scatterplot.svghttp://en.wikipedia.org/wiki/Perceptronhttp://en.wikipedia.org/wiki/Logistic_regressionhttp://en.wikipedia.org/wiki/Artificial_neural_networkhttp://en.wikipedia.org/wiki/Linear_regressionhttp://en.wikipedia.org/wiki/K-nearest_neighbors_classificationhttp://en.wikipedia.org/wiki/Random_foresthttp://en.wikipedia.org/wiki/Boosting_%28machine_learning%29http://en.wikipedia.org/wiki/Bootstrap_aggregatinghttp://en.wikipedia.org/wiki/Ensemble_learninghttp://en.wikipedia.org/wiki/Decision_tree_learninghttp://en.wikipedia.org/wiki/Regression_analysishttp://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/wiki/Supervised_learninghttp://en.wikipedia.org/wiki/Grammar_inductionhttp://en.wikipedia.org/wiki/Semi-supervised_learninghttp://en.wikipedia.org/wiki/Online_machine_learninghttp://en.wikipedia.org/wiki/Feature_learninghttp://en.wikipedia.org/wiki/Structured_predictionhttp://en.wikipedia.org/wiki/Reinforcement_learninghttp://en.wikipedia.org/wiki/Association_rule_learninghttp://en.wikipedia.org/wiki/Anomaly_detectionhttp://en.wikipedia.org/wiki/Regression_analysishttp://en.wikipedia.org/wiki/Cluster_analysishttp://en.wikipedia.org/wiki/Statistical_classificationhttp://en.wikipedia.org/wiki/Data_mininghttp://en.wikipedia.org/wiki/Machine_learning


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Support vector machine (SVM)

Relevance vector machine (RVM)

Clustering

BIRCH

Hierarchical

k-means

Expectation-maximization (EM)

DBSCAN

OPTICS

Mean-shift

Dimensionality reduction

Factor analysis

CCA

ICA

LDA

NMF

PCA

t-SNE

Structured prediction

Graphical models(Bayes net,CRF,HMM)

Anomaly detection

k-NN

Local outlier factor

Neural nets

Autoencoder

Deep learning

Multilayer perceptron

RNN

Restricted Boltzmann machine

SOM
http://en.wikipedia.org/wiki/Support_vector_machinehttp://en.wikipedia.org/wiki/Support_vector_machinehttp://en.wikipedia.org/wiki/Relevance_vector_machinehttp://en.wikipedia.org/wiki/Relevance_vector_machinehttp://en.wikipedia.org/wiki/Cluster_analysishttp://en.wikipedia.org/wiki/BIRCH_%28data_clustering%29http://en.wikipedia.org/wiki/BIRCH_%28data_clustering%29http://en.wikipedia.org/wiki/Hierarchical_clusteringhttp://en.wikipedia.org/wiki/Hierarchical_clusteringhttp://en.wikipedia.org/wiki/K-means_clusteringhttp://en.wikipedia.org/wiki/K-means_clusteringhttp://en.wikipedia.org/wiki/K-means_clusteringhttp://en.wikipedia.org/wiki/Expectation-maximization_algorithmhttp://en.wikipedia.org/wiki/Expectation-maximization_algorithmhttp://en.wikipedia.org/wiki/DBSCANhttp://en.wikipedia.org/wiki/DBSCANhttp://en.wikipedia.org/wiki/OPTICS_algorithmhttp://en.wikipedia.org/wiki/OPTICS_algorithmhttp://en.wikipedia.org/wiki/Mean-shifthttp://en.wikipedia.org/wiki/Mean-shifthttp://en.wikipedia.org/wiki/Dimensionality_reductionhttp://en.wikipedia.org/wiki/Factor_analysishttp://en.wikipedia.org/wiki/Factor_analysishttp://en.wikipedia.org/wiki/Canonical_correlation_analysishttp://en.wikipedia.org/wiki/Canonical_correlation_analysishttp://en.wikipedia.org/wiki/Independent_component_analysishttp://en.wikipedia.org/wiki/Independent_component_analysishttp://en.wikipedia.org/wiki/Linear_discriminant_analysishttp://en.wikipedia.org/wiki/Linear_discriminant_analysishttp://en.wikipedia.org/wiki/Non-negative_matrix_factorizationhttp://en.wikipedia.org/wiki/Non-negative_matrix_factorizationhttp://en.wikipedia.org/wiki/Principal_component_analysishttp://en.wikipedia.org/wiki/Principal_component_analysishttp://en.wikipedia.org/wiki/T-distributed_stochastic_neighbor_embeddinghttp://en.wikipedia.org/wiki/T-distributed_stochastic_neighbor_embeddinghttp://en.wikipedia.org/wiki/Structured_predictionhttp://en.wikipedia.org/wiki/Graphical_modelhttp://en.wikipedia.org/wiki/Graphical_modelhttp://en.wikipedia.org/wiki/Bayesian_networkhttp://en.wikipedia.org/wiki/Bayesian_networkhttp://en.wikipedia.org/wiki/Bayesian_networkhttp://en.wikipedia.org/wiki/Conditional_random_fieldhttp://en.wikipedia.org/wiki/Conditional_random_fieldhttp://en.wikipedia.org/wiki/Conditional_random_fieldhttp://en.wikipedia.org/wiki/Hidden_Markov_modelhttp://en.wikipedia.org/wiki/Hidden_Markov_modelhttp://en.wikipedia.org/wiki/Hidden_Markov_modelhttp://en.wikipedia.org/wiki/Anomaly_detectionhttp://en.wikipedia.org/wiki/K-nearest_neighbors_classificationhttp://en.wikipedia.org/wiki/K-nearest_neighbors_classificationhttp://en.wikipedia.org/wiki/K-nearest_neighbors_classificationhttp://en.wikipedia.org/wiki/Local_outlier_factorhttp://en.wikipedia.org/wiki/Local_outlier_factorhttp://en.wikipedia.org/wiki/Artificial_neural_networkhttp://en.wikipedia.org/wiki/Autoencoderhttp://en.wikipedia.org/wiki/Autoencoderhttp://en.wikipedia.org/wiki/Deep_learninghttp://en.wikipedia.org/wiki/Deep_learninghttp://en.wikipedia.org/wiki/Multilayer_perceptronhttp://en.wikipedia.org/wiki/Multilayer_perceptronhttp://en.wikipedia.org/wiki/Recurrent_neural_networkhttp://en.wikipedia.org/wiki/Recurrent_neural_networkhttp://en.wikipedia.org/wiki/Restricted_Boltzmann_machinehttp://en.wikipedia.org/wiki/Restricted_Boltzmann_machinehttp://en.wikipedia.org/wiki/Self-organizing_maphttp://en.wikipedia.org/wiki/Self-organizing_maphttp://en.wikipedia.org/wiki/Self-organizing_maphttp://en.wikipedia.org/wiki/Restricted_Boltzmann_machinehttp://en.wikipedia.org/wiki/Recurrent_neural_networkhttp://en.wikipedia.org/wiki/Multilayer_perceptronhttp://en.wikipedia.org/wiki/Deep_learninghttp://en.wikipedia.org/wiki/Autoencoderhttp://en.wikipedia.org/wiki/Artificial_neural_networkhttp://en.wikipedia.org/wiki/Local_outlier_factorhttp://en.wikipedia.org/wiki/K-nearest_neighbors_classificationhttp://en.wikipedia.org/wiki/Anomaly_detectionhttp://en.wikipedia.org/wiki/Hidden_Markov_modelhttp://en.wikipedia.org/wiki/Conditional_random_fieldhttp://en.wikipedia.org/wiki/Bayesian_networkhttp://en.wikipedia.org/wiki/Graphical_modelhttp://en.wikipedia.org/wiki/Structured_predictionhttp://en.wikipedia.org/wiki/T-distributed_stochastic_neighbor_embeddinghttp://en.wikipedia.org/wiki/Principal_component_analysishttp://en.wikipedia.org/wiki/Non-negative_matrix_factorizationhttp://en.wikipedia.org/wiki/Linear_discriminant_analysishttp://en.wikipedia.org/wiki/Independent_component_analysishttp://en.wikipedia.org/wiki/Canonical_correlation_analysishttp://en.wikipedia.org/wiki/Factor_analysishttp://en.wikipedia.org/wiki/Dimensionality_reductionhttp://en.wikipedia.org/wiki/Mean-shifthttp://en.wikipedia.org/wiki/OPTICS_algorithmhttp://en.wikipedia.org/wiki/DBSCANhttp://en.wikipedia.org/wiki/Expectation-maximization_algorithmhttp://en.wikipedia.org/wiki/K-means_clusteringhttp://en.wikipedia.org/wiki/Hierarchical_clusteringhttp://en.wikipedia.org/wiki/BIRCH_%28data_clustering%29http://en.wikipedia.org/wiki/Cluster_analysishttp://en.wikipedia.org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Convolutional neural network

Theory

Bias-variance dilemma

Computational learning theory

Empirical risk minimization

PAC learning

Statistical learning

VC theory

Computer science portal

Statistics portal

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Inmachine learning,naive Bayes classifiersare a family of simpleprobabilistic classifiers

based on applyingBayes' theoremwith strong (naive)independenceassumptions between the

features.

Naive Bayes models are also known under a variety of names in the literature, including

simple Bayesand independence Bayes.[1]All these names reference the use of Bayes'

theorem in the classifier's decision rule, but naive Bayes is not (necessarily) aBayesian

method;[1]Russell and Norvignote that "[naive Bayes] is sometimes called a Bayesian

classifier, a somewhat careless usage that has prompted true Bayesians to call it the idiot

Bayesmodel."[2]:482

Naive Bayes has been studied extensively since the 1950s. It was introduced under a different

name into thetext retrievalcommunity in the early 1960s,[2]:488and remains a popular

(baseline) method fortext categorization,the problem of judging documents as belonging toone category or the other (such asspam or legitimate,sports or politics, etc.) withword

frequenciesas the features. With appropriate preprocessing, it is competitive in this domain

with more advanced methods includingsupport vector machines.[3]It also finds application in

automaticmedical diagnosis.[4]

Naive Bayes classifiers are highly scalable, requiring a number of parameters linear in the

number of variables (features/predictors) in a learning problem.Maximum-likelihoodtraining

can be done by evaluating aclosed-form expression,[2]:718which takeslinear time,rather than

by expensiveiterative approximationas used for many other types of classifiers.

Contents
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tive_methodhttp://en.wikipedia.org/wiki/Iterative_methodhttp://en.wikipedia.org/wiki/File:Fisher_iris_versicolor_sepalwidth.svghttp://en.wikipedia.org/wiki/File:Internet_map_1024.jpghttp://en.wikipedia.org/wiki/File:Fisher_iris_versicolor_sepalwidth.svghttp://en.wikipedia.org/wiki/File:Internet_map_1024.jpghttp://en.wikipedia.org/wiki/Iterative_methodhttp://en.wikipedia.org/wiki/Linear_timehttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-aima-2http://en.wikipedia.org/wiki/Closed-form_expressionhttp://en.wikipedia.org/wiki/Maximum-likelihood_estimationhttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-rish-4http://en.wikipedia.org/wiki/Medical_diagnosishttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-rennie-3http://en.wikipedia.org/wiki/Support_vector_machinehttp://en.wikipedia.org/wiki/Bag_of_wordshttp://en.wikipedia.org/wiki/Bag_of_wordshttp://en.wikipedia.org/wiki/Spam_filteringhttp://en.wikipedia.org/wiki/Text_categorizationhttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-aima-2http://en.wikipedia.org/wiki/Information_retrievalhttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-aima-2http://en.wikipedia.org/wiki/Artificial_Intelligence:_A_Modern_Approachhttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-idiots-1http://en.wikipedia.org/wiki/Bayesian_probabilityhttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-idiots-1http://en.wikipedia.org/wiki/Statistical_independencehttp://en.wikipedia.org/wiki/Bayes%27_theoremhttp://en.wikipedia.org/wiki/Probabilistic_classifierhttp://en.wikipedia.org/wiki/Machine_learninghttp://en.wikipedia.org/w/index.php?title=Template:Machine_learning_bar&action=edithttp://en.wikipedia.org/wiki/Template_talk:Machine_learning_barhttp://en.wikipedia.org/wiki/Template:Machine_learning_barhttp://en.wikipedia.org/wiki/Portal:Statisticshttp://en.wikipedia.org/wiki/Portal:Computer_sciencehttp://en.wikipedia.org/wiki/Vapnik%E2%80%93Chervonenkis_theoryhttp://en.wikipedia.org/wiki/Statistical_learning_theoryhttp://en.wikipedia.org/wiki/Probably_approximately_correct_learninghttp://en.wikipedia.org/wiki/Empirical_risk_minimizationhttp://en.wikipedia.org/wiki/Computational_learning_theoryhttp://en.wikipedia.org/wiki/Bias-variance_dilemmahttp://en.wikipedia.org/wiki/Convolutional_neural_network


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1 Introduction

2 Probabilistic model

o 2.1 Constructing a classifier from the probability model

3 Parameter estimation and event models

o 3.1 Gaussian naive Bayes

o

3.2 Multinomial naive Bayeso 3.3 Bernoulli naive Bayes

o 3.4 Semi-supervised parameter estimation

4 Discussion

o 4.1 Relation to logistic regression

5 Examples

o 5.1 Sex classification

5.1.1 Training

5.1.2 Testing

o 5.2 Document classification

6 See also

7 Referenceso 7.1 Further reading

8 External links

Introduction

In simple terms, a naive Bayes classifier assumes that the value of a particular feature is

unrelated to the presence or absence of any other feature, given the class variable. For

example, a fruit may be considered to be an apple if it is red, round, and about 3" in diameter.

A naive Bayes classifier considers each of these features to contribute independently to the

probability that this fruit is an apple, regardless of the presence or absence of the otherfeatures.

For some types of probability models, naive Bayes classifiers can be trained very efficiently

in asupervised learningsetting. In many practical applications, parameter estimation for

naive Bayes models uses the method ofmaximum likelihood;in other words, one can work

with the naive Bayes model without acceptingBayesian probabilityor using any Bayesian

methods.

Despite their naive design and apparently oversimplified assumptions, naive Bayes classifiers

have worked quite well in many complex real-world situations. In 2004, an analysis of the

Bayesian classification problem showed that there are sound theoretical reasons for theapparently implausibleefficacyof naive Bayes classifiers.[5]Still, a comprehensive

comparison with other classification algorithms in 2006 showed that Bayes classification is

outperformed by other approaches, such asboosted treesorrandom forests.[6]

An advantage of naive Bayes is that it only requires a small amount of training data to

estimate the parameters (means and variances of the variables) necessary for classification.

Because independent variables are assumed, only the variances of the variables for each class

need to be determined and not the entirecovariance matrix.

Probabilistic model
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Abstractly, the probability model for a classifier is a conditional model

over a dependent class variable with a small number of outcomes or classes, conditional on

several feature variables through . The problem is that if the number of features islarge or if a feature can take on a large number of values, then basing such a model on

probability tables is infeasible. We therefore reformulate the model to make it more tractable.

UsingBayes' theorem,this can be written

In plain English, usingBayesian Probabilityterminology, the above equation can be written

as

In practice, there is interest only in the numerator of that fraction, because the denominator

does not depend on and the values of the features are given, so that the denominator is

effectively constant. The numerator is equivalent to thejoint probabilitymodel

which can be rewritten as follows, using thechain rulefor repeated applications of the

definition ofconditional probability:

Now the "naive"conditional independenceassumptions come into play: assume that each

feature is conditionallyindependentof every other feature for , given the

category . This means that

,

,

,

and so on, for . Thus, the joint model can be expressed as
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This means that under the above independence assumptions, the conditional distribution over

the class variable is:

where the evidence is a scaling factor dependent only on

, that is, a constant if the values of the feature variables are known.

Constructing a classifier from the probability model

The discussion so far has derived the independent feature model, that is, the naive Bayes

probability model.The naive Bayesclassifiercombines this model with adecision rule.One

common rule is to pick the hypothesis that is most probable; this is known as themaximum a

posterioriorMAPdecision rule. The corresponding classifier, aBayes classifier,is the

function defined as follows:

Parameter estimation and event models

A class' prior may be calculated by assuming equiprobable classes (i.e., priors = 1 / (number

of classes)), or by calculating an estimate for the class probability from the training set (i.e.,

(prior for a given class) = (number of samples in the class) / (total number of samples)). To

estimate the parameters for a feature's distribution, one must assume a distribution or

generatenonparametricmodels for the features from the training set.[7]

The assumptions on distributions of features are called the event modelof the Naive Bayes

classifier. For discrete features like the ones encountered in document classification (include

spam filtering),multinomialandBernoullidistributions are popular. These assumptions lead

to two distinct models, which are often confused.[8][9]

Gaussian naive Bayes

When dealing with continuous data, a typical assumption is that the continuous values

associated with each class are distributed according to aGaussiandistribution. For example,

suppose the training data contain a continuous attribute, . We first segment the data by the

class, and then compute the mean andvarianceof in each class. Let be the mean of the
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values in associated with class c, and let be the variance of the values in associated

with class c. Then, the probability densityof some value given a class, , can be

computed by plugging into the equation for aNormal distributionparameterized by and

. That is,

Another common technique for handling continuous values is to use binning todiscretizethe

feature values, to obtain a new set of Bernoulli-distributed features; some literature in fact

suggests that this is necessary to apply naive Bayes, but it is not, and the discretization may

throw away discriminative information.[1]

Multinomial naive Bayes

With a multinomial event model, samples (feature vectors) represent the frequencies with

which certain events have been generated by amultinomial where is the

probability that event ioccurs (or ksuch multinomials in the multiclass case). This is the

event model typically used for document classification; the feature values are then term

frequencies, generated by a multinomial that produces some number of words (seebag of

wordsassumption). The likelihood of observing a feature vector (histogram)Fis given by

The multinomial naive Bayes classifier becomes alinear classifierwhen expressed in log-

space:[3]

where and .

If a given class and feature value never occur together in the training data, then the

frequency-based probability estimate will be zero. This is problematic because it will wipe

out all information in the other probabilities when they are multiplied. Therefore, it is often

desirable to incorporate a small-sample correction, calledpseudocount,in all probability

estimates such that no probability is ever set to be exactly zero. This way ofregularizing

naive Bayes is calledLaplace smoothingwhen the pseudocount is one, andLidstonesmoothingin the general case.
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Rennie et al.discuss problems with the multinomial assumption in the context of document

classification and possible ways to alleviate those problems, including the use oftfidf

weights instead of raw term frequencies and document length normalization, to produce a

naive Bayes classifier that is competitive withsupport vector machines.[3]

Bernoulli naive Bayes

In the multivariateBernoullievent model, features are independentbooleans(binary

variables) describing inputs. This model is also popular for document classification tasks,[8]

where binary term occurrence features are used rather than term frequencies. If is a

boolean expressing the occurrence or absence of the i'th term from the vocabulary, then the

likelihood of a document given a class Cis given by[8]

where is the probability of class Cgenerating the term . This event model is

especially popular for classifying short texts. It has the benefit of explicitly modelling the

absence of terms. Note that a naive Bayes classifier with a Bernoulli event model is not the

same as a multinomial NB classifier with frequency counts truncated to one.

Semi-supervised parameter estimation

Given a way to train a naive Bayes classifier from labeled data, it's possible to construct a

semi-supervisedtraining algorithm that can learn from a combination of labeled and

unlabeled data by running the supervised learning algorithm in a loop:[10]

Given a collection of labeled samplesLand unlabeled samples U, start

by training a naive Bayes classifier onL.

Until convergence, do:

Predict class probabilities for all examplesxin .

Re-train the model based on theprobabilities(not the labels) predicted in the previous

step.

Convergence is determined based on improvement to the model likelihood , where

denotes the parameters of the naive Bayes model.

This training algorithm is an instance of the more generalexpectationmaximization

algorithm(EM): the prediction step inside the loop is theE-step of EM, while the re-training

of naive Bayes is theM-step. The algorithm is formally justified by the assumption that the

data are generated by amixture model,and the components of this mixture model are exactly

the classes of the classification problem.[10]

Discussion

Despite the fact that the far-reaching independence assumptions are often inaccurate, thenaive Bayes classifier has several properties that make it surprisingly useful in practice. In
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particular, the decoupling of the class conditional feature distributions means that each

distribution can be independently estimated as a one-dimensional distribution. This helps

alleviate problems stemming from thecurse of dimensionality,such as the need for data sets

that scale exponentially with the number of features. While naive Bayes often fails to produce

a good estimate for the correct class probabilities,[11]this may not be a requirement for many

applications. For example, the naive Bayes classifier will make the correct MAP decision ruleclassification so long as the correct class is more probable than any other class. This is true

regardless of whether the probability estimate is slightly, or even grossly inaccurate. In this

manner, the overall classifier can be robust enough to ignore serious deficiencies in its

underlying naive probability model.[4]Other reasons for the observed success of the naive

Bayes classifier are discussed in the literature cited below.

Relation to logistic regression

This section requiresexpansion.(August 2014)

In the case of discrete inputs (indicator or frequency features for discrete events), naive Bayes

classifiers form agenerative-discriminativepair with (multinomial)logistic regression

classifiers: each naive Bayes classifier can be considered a way of fitting a probability model

p(x,y) to optimize the joint likelihoodp(x,y), while logistic regression fits the same

probability model to optimize the conditionalp(x|y).[12]

The link between the two can be seen by observing that the decision function for naive Bayes

(in binary classification) can be rewritten as "predict 1 if theoddsofp(y=1 |x) are higher than

those ofp(y=0 |x)". Expressing this in log-space gives:

The left-hand side of this equation is the log-odds, orlogit,the quantity predicted by the

linear model that underlies logistic regression. Since naive Bayes is also a linear model in the

discrete case, it can be be reparametrised as a linear function . Obtaining the

probabilities is then a matter of applying thelogistic functionto , or in themulticlass case, thesoftmax function.

Discriminative classifiers have lower asymptotic error than generative ones; however,research byNgandJordanhas shown that in some practical cases naive Bayes can

outperform logistic regression because it reaches its asymptotic error faster.[12]

Examples

Sex classification

Problem: classify whether a given person is a male or a female based on the measured

features. The features include height, weight, and foot size.

Training
http://en.wikipedia.org/wiki/Curse_of_dimensionalityhttp://en.wikipedia.org/wiki/Curse_of_dimensionalityhttp://en.wikipedia.org/wiki/Curse_of_dimensionalityhttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-11http://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-11http://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-11http://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-rish-4http://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-rish-4http://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-rish-4http://en.wikipedia.org/w/index.php?title=Naive_Bayes_classifier&action=edithttp://en.wikipedia.org/w/index.php?title=Naive_Bayes_classifier&action=edithttp://en.wikipedia.org/w/index.php?title=Naive_Bayes_classifier&action=edithttp://en.wikipedia.org/wiki/Multinomial_logistic_regressionhttp://en.wikipedia.org/wiki/Multinomial_logistic_regressionhttp://en.wikipedia.org/wiki/Multinomial_logistic_regressionhttp://en.wikipedia.org/wiki/Logistic_regressionhttp://en.wikipedia.org/wiki/Logistic_regressionhttp://en.wikipedia.org/wiki/Logistic_regressionhttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-pair-12http://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-pair-12http://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-pair-12http://en.wikipedia.org/wiki/Oddshttp://en.wikipedia.org/wiki/Oddshttp://en.wikipedia.org/wiki/Oddshttp://en.wikipedia.org/wiki/Logithttp://en.wikipedia.org/wiki/Logithttp://en.wikipedia.org/wiki/Logithttp://en.wikipedia.org/wiki/Logistic_functionhttp://en.wikipedia.org/wiki/Logistic_functionhttp://en.wikipedia.org/wiki/Logistic_functionhttp://en.wikipedia.org/wiki/Softmax_functionhttp://en.wikipedia.org/wiki/Softmax_functionhttp://en.wikipedia.org/wiki/Softmax_functionhttp://en.wikipedia.org/wiki/Andrew_Nghttp://en.wikipedia.org/wiki/Andrew_Nghttp://en.wikipedia.org/wiki/Andrew_Nghttp://en.wikipedia.org/wiki/Michael_I._Jordanhttp://en.wikipedia.org/wiki/Michael_I._Jordanhttp://en.wikipedia.org/wiki/Michael_I._Jordanhttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-pair-12http://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-pair-12http://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-pair-12http://en.wikipedia.org/wiki/File:Wiki_letter_w_cropped.svghttp://en.wikipedia.org/wiki/File:Wiki_letter_w_cropped.svghttp://en.wikipedia.org/wiki/File:Wiki_letter_w_cropped.svghttp://en.wikipedia.org/wiki/File:Wiki_letter_w_cropped.svghttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-pair-12http://en.wikipedia.org/wiki/Michael_I._Jordanhttp://en.wikipedia.org/wiki/Andrew_Nghttp://en.wikipedia.org/wiki/Softmax_functionhttp://en.wikipedia.org/wiki/Logistic_functionhttp://en.wikipedia.org/wiki/Logithttp://en.wikipedia.org/wiki/Oddshttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-pair-12http://en.wikipedia.org/wiki/Logistic_regressionhttp://en.wikipedia.org/wiki/Multinomial_logistic_regressionhttp://en.wikipedia.org/w/index.php?title=Naive_Bayes_classifier&action=edithttp://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-rish-4http://en.wikipedia.org/wiki/Naive_Bayes_classifier#cite_note-11http://en.wikipedia.org/wiki/Curse_of_dimensionality


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Example training set below.

sex height (feet) weight (lbs) foot size(inches)

male 6 180 12

male 5.92 (5'11") 190 11male 5.58 (5'7") 170 12

male 5.92 (5'11") 165 10

female 5 100 6

female 5.5 (5'6") 150 8

female 5.42 (5'5") 130 7

female 5.75 (5'9") 150 9

The classifier created from the training set using a Gaussian distribution assumption would be

(given variances aresample variances):

sexmean

(height)

variance

(height)

mean

(weight)

variance

(weight)

mean (foot

size)

variance (foot

size)

male 5.855 3.5033e-02 176.25 1.2292e+02 11.25 9.1667e-01

female 5.4175 9.7225e-02 132.5 5.5833e+02 7.5 1.6667e+00

Let's say we have equiprobable classes so P(male)= P(female) = 0.5. This prior probability

distribution might be based on our knowledge of frequencies in the larger population, or on

frequency in the training set.

Testing

Below is a sample to be classified as a male or female.

sex height (feet) weight (lbs) foot size(inches)

sample 6 130 8

We wish to determine which posterior is greater, male or female. For the classification as

male the posterior is given by

For the classification as female the posterior is given by

The evidence (also termed normalizing constant) may be calculated:
http://en.wikipedia.org/wiki/Variance#Population_variance_and_sample_variancehttp://en.wikipedia.org/wiki/Variance#Population_variance_and_sample_variancehttp://en.wikipedia.org/wiki/Variance#Population_variance_and_sample_variancehttp://en.wikipedia.org/wiki/Variance#Population_variance_and_sample_variance


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However, given the sample the evidence is a constant and thus scales both posteriors equally.

It therefore does not affect classification and can be ignored. We now determine the

probability distribution for the sex of the sample.

,

where and are the parameters of normal distribution

which have been previously determined from the training set. Note that a value greater than 1

is OK hereit is a probability density rather than a probability, because height is a

continuous variable.

Since posterior numerator is greater in the female case, we predict the sample is female.

Document classification

Here is a worked example of naive Bayesian classification to thedocument classification

problem. Consider the problem of classifying documents by their content, for example into

spamand non-spame-mails.Imagine that documents are drawn from a number of classes of

documents which can be modelled as sets of words where the (independent) probability that

the i-th word of a given document occurs in a document from class Ccan be written as

(For this treatment, we simplify things further by assuming that words are randomlydistributed in the document - that is, words are not dependent on the length of the document,

position within the document with relation to other words, or other document-context.)

Then the probability that a given documentDcontains all of the words , given a class C, is

The question that we desire to answer is: "what is the probability that a given documentD

belongs to a given class C?" In other words, what is ?
http://en.wikipedia.org/wiki/Document_classificationhttp://en.wikipedia.org/wiki/Document_classificationhttp://en.wikipedia.org/wiki/Document_classificationhttp://en.wikipedia.org/wiki/Spamminghttp://en.wikipedia.org/wiki/Spamminghttp://en.wikipedia.org/wiki/E-mailhttp://en.wikipedia.org/wiki/E-mailhttp://en.wikipedia.org/wiki/E-mailhttp://en.wikipedia.org/wiki/E-mailhttp://en.wikipedia.org/wiki/Spamminghttp://en.wikipedia.org/wiki/Document_classification


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Nowby definition

and

Bayes' theorem manipulates these into a statement of probability in terms oflikelihood.

Assume for the moment that there are only two mutually exclusive classes, Sand S(e.g.

spam and not spam), such that every element (email) is in either one or the other;

and

Using the Bayesian result above, we can write:

Dividing one by the other gives:

Which can be re-factored as:
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Thus, the probability ratio p(S|D) / p(S|D) can be expressed in terms of a series of

likelihood ratios.The actual probability p(S|D) can be easily computed from log (p(S|D) /

p(S|D)) based on the observation that p(S|D) + p(S|D) = 1.

Taking thelogarithmof all these ratios, we have:

(This technique of "log-likelihood ratios" is a common technique in statistics. In the case oftwo mutually exclusive alternatives (such as this example), the conversion of a log-likelihood

ratio to a probability takes the form of asigmoid curve:seelogitfor details.)

Finally, the document can be classified as follows. It is spam if (i.e.,

), otherwise it is not spam.
http://en.wikipedia.org/wiki/Likelihood_functionhttp://en.wikipedia.org/wiki/Likelihood_functionhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Log-likelihood_ratiohttp://en.wikipedia.org/wiki/Log-likelihood_ratiohttp://en.wikipedia.org/wiki/Sigmoid_curvehttp://en.wikipedia.org/wiki/Sigmoid_curvehttp://en.wikipedia.org/wiki/Sigmoid_curvehttp://en.wikipedia.org/wiki/Logithttp://en.wikipedia.org/wiki/Logithttp://en.wikipedia.org/wiki/Logithttp://en.wikipedia.org/wiki/Logithttp://en.wikipedia.org/wiki/Sigmoid_curvehttp://en.wikipedia.org/wiki/Log-likelihood_ratiohttp://en.wikipedia.org/wiki/Logarithmhttp://en.wikipedia.org/wiki/Likelihood_function

Naive Bayes Classifier From Wikipedia

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