In machine learning, naive Bayes classifiers are a family of simple probabilistic classifiers based on applying Bayes' theorem with strong (naive) independence assumptions between the features.
Naive Bayes has been studied extensively since the 1950s. It was introduced under a different name into the text retrieval community in the early 1960s,[1]:488 and remains a popular (baseline) method for text categorization, the problem of judging documents as belonging to one category or the other (such as spam or legitimate, sports or politics, etc.) with word frequencies as the features. With appropriate preprocessing, it is competitive in this domain with more advanced methods including support vector machines.[2] It also finds application in automatic medical diagnosis.[3]
Naive Bayes classifiers are highly scalable, requiring a number of parameters linear in the number of variables (features/predictors) in a learning problem. Maximumlikelihood training can be done by evaluating a closedform expression,[1]:718 which takes linear time, rather than by expensive iterative approximation as used for many other types of classifiers.
In the statistics and computer science literature, Naive Bayes models are known under a variety of names, including simple Bayes and independence Bayes.[4] All these names reference the use of Bayes' theorem in the classifier's decision rule, but naive Bayes is not (necessarily) a Bayesian method;[4] Russell and Norvig note that "[naive Bayes] is sometimes called a Bayesian classifier, a somewhat careless usage that has prompted true Bayesians to call it the idiot Bayes model."[1]:482
Introduction
Naive Bayes is a simple technique for constructing classifiers: models that assign class labels to problem instances, represented as vectors of feature values, where the class labels are drawn from some finite set. It is not a single algorithm for training such classifiers, but a family of algorithms based on a common principle: all naive Bayes classifiers assume that the value of a particular feature is independent of the value 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 any possible correlations between the color, roundness and diameter features.
For some types of probability models, naive Bayes classifiers can be trained very efficiently in a supervised learning setting. In many practical applications, parameter estimation for naive Bayes models uses the method of maximum likelihood; in other words, one can work with the naive Bayes model without accepting Bayesian probability or using any Bayesian methods.
Despite their naive design and apparently oversimplified assumptions, naive Bayes classifiers have worked quite well in many complex realworld situations. In 2004, an analysis of the Bayesian classification problem showed that there are sound theoretical reasons for the apparently implausible efficacy of naive Bayes classifiers.[5] Still, a comprehensive comparison with other classification algorithms in 2006 showed that Bayes classification is outperformed by other approaches, such as boosted trees or random forests.[6]
An advantage of naive Bayes is that it only requires a small amount of training data to estimate the parameters necessary for classification.[citation needed]
Probabilistic model
Abstractly, naive Bayes is a conditional probability model: given a problem instance to be classified, represented by a vector
representing some n features (dependent variables), it assigns to this instance probabilities
for each of k possible outcomes or classes.[7]
The problem with the above formulation is that if the number of features n is large 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. Using Bayes' theorem, the conditional probability can be decomposed as
In plain English, using Bayesian probability terminology, 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 C and the values of the features F_{i} are given, so that the denominator is effectively constant. The numerator is equivalent to the joint probability model
which can be rewritten as follows, using the chain rule for repeated applications of the definition of conditional probability:
Now the "naive" conditional independence assumptions come into play: assume that each feature F
_{i} is conditionally independent of every other feature F
_{j} for
, given the category C. This means that
and so on, for
. Thus, the joint model can be expressed as
This means that under the above independence assumptions, the conditional distribution over the class variable C 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 Bayes classifier combines this model with a decision rule. One common rule is to pick the hypothesis that is most probable; this is known as the maximum a posteriori or MAP decision rule. The corresponding classifier, a Bayes classifier, is the function that assigns a class
for some k 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 generate nonparametric models for the features from the training set.[8]
The assumptions on distributions of features are called the event model of the Naive Bayes classifier. For discrete features like the ones encountered in document classification (include spam filtering), multinomial and Bernoulli distributions are popular. These assumptions lead to two distinct models, which are often confused.[9][10]
Gaussian naive Bayes
When dealing with continuous data, a typical assumption is that the continuous values associated with each class are distributed according to a Gaussian distribution. For example, suppose the training data contain a continuous attribute, x. We first segment the data by the class, and then compute the mean and variance of x in each class. Let
be the mean of the values in x associated with class c, and let
be the variance of the values in x associated with class c. Then, the probability distribution of some value given a class, p(x=vc), can be computed by plugging v into the equation for a Normal distribution parameterized by
and
. That is,
Another common technique for handling continuous values is to use binning to discretize the feature values, to obtain a new set of Bernoullidistributed 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.[4]
Multinomial naive Bayes
With a multinomial event model, samples (feature vectors) represent the frequencies with which certain events have been generated by a multinomial
where
is the probability that event i occurs (or K such multinomials in the multiclass case). A feature vector
is then a histogram, with X
_{i} counting the number of times event i was observed in a particular instance. This is the event model typically used for document classification, with events representing the occurrence of a word in a single document (see bag of words assumption). The likelihood of observing a histogram x is given by
The multinomial naive Bayes classifier becomes a linear classifier when expressed in logspace:[2]
where
If a given class and feature value never occur together in the training data, then the frequencybased 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 smallsample correction, called pseudocount, in all probability estimates such that no probability is ever set to be exactly zero. This way of regularizing naive Bayes is called Laplace smoothing when the pseudocount is one, and Lidstone smoothing in the general case.
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 of tf–idf weights instead of raw term frequencies and document length normalization, to produce a naive Bayes classifier that is competitive with support vector machines.[2]
Bernoulli naive Bayes
In the multivariate Bernoulli event model, features are independent booleans (binary variables) describing inputs. Like the multinomial model, this model is popular for document classification tasks,[9] where binary term occurrence features are used rather than term frequencies. If X_{i} 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 C_{k} is given by[9]
where
is the probability of class C
_{k} generating the term w
_{i}. 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.
Semisupervised parameter estimation
Given a way to train a naive Bayes classifier from labeled data, it's possible to construct a semisupervised training algorithm that can learn from a combination of labeled and unlabeled data by running the supervised learning algorithm in a loop:[11]
Given a collection
of labeled samples L and unlabeled samples U, start by training a naive Bayes classifier on L.
Until convergence, do:
Predict class probabilities P(Cx) for all examples x in D.
Retrain the model based on the probabilities (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 general expectation–maximization algorithm (EM): the prediction step inside the loop is the Estep of EM, while the retraining of naive Bayes is the Mstep. The algorithm is formally justified by the assumption that the data are generated by a mixture model, and the components of this mixture model are exactly the classes of the classification problem.[11]
Discussion
Despite the fact that the farreaching independence assumptions are often inaccurate, the naive Bayes classifier has several properties that make it surprisingly useful in practice. In particular, the decoupling of the class conditional feature distributions means that each distribution can be independently estimated as a onedimensional distribution. This helps alleviate problems stemming from the curse 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,[12] this may not be a requirement for many applications. For example, the naive Bayes classifier will make the correct MAP decision rule classification 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.[3] Other reasons for the observed success of the naive Bayes classifier are discussed in the literature cited below.
References

Russell, Stuart; Norvig, Peter (2003) [1995]. Artificial Intelligence: A Modern Approach (2nd ed.). Prentice Hall. ISBN 9780137903955.

Rennie, J.; Shih, L.; Teevan, J.; Karger, D. (2003). Tackling the poor assumptions of Naive Bayes classifiers

Rish, Irina (2001). An empirical study of the naive Bayes classifier . IJCAI Workshop on Empirical Methods in AI.

Hand, D. J.; Yu, K. (2001). "Idiot's Bayes — not so stupid after all?". International Statistical Review 69 (3): 385–399. doi:10.2307/1403452. ISSN 03067734

Zhang, Harry. The Optimality of Naive Bayes . FLAIRS2004 conference.

Caruana, R.; NiculescuMizil, A. (2006). An empirical comparison of supervised learning algorithms. Proc. 23rd International Conference on Machine Learning. CiteSeerX: 10.1.1.122.5901

Narasimha Murty, M.; Susheela Devi, V. (2011). Pattern Recognition: An Algorithmic Approach. ISBN 0857294946.

ohn, George H.; Langley, Pat (1995). Estimating Continuous Distributions in Bayesian Classifiers. Proc. Eleventh Conf. on Uncertainty in Artificial Intelligence. Morgan Kaufmann. pp. 338–345.

McCallum, Andrew; Nigam, Kamal (1998). A comparison of event models for Naive Bayes text classification . AAAI98 workshop on learning for text categorization 752.

Metsis, Vangelis; Androutsopoulos, Ion; Paliouras, Georgios (2006). Spam filtering with Naive Bayes—which Naive Bayes?. Third conference on email and antispam (CEAS) 17.

Nigam, Kamal; McCallum, Andrew; Thrun, Sebastian; Mitchell, Tom (2000). "Learning to classify text from labeled and unlabeled documents using EM" . Machine Learning.

NiculescuMizil, Alexandru; Caruana, Rich (2005). Predicting good probabilities with supervised learning . ICML. doi:10.1145/1102351.1102430
* https://en.wikipedia.org/