Naive Bayes classifier

thumb|Example of a naive Bayes classifier depicted as a Bayesian network

In statistics, naive (sometimes simple or idiot's) Bayes classifiers are a family of "probabilistic classifiers" which assume that the features are conditionally independent, given the target class. In other words, a naive Bayes model assumes the information about the class provided by each variable is unrelated to the information from the others, with no information shared between the predictors. The highly unrealistic nature of this assumption, called the naive independence assumption, is what gives the classifier its name. These classifiers are some of the simplest Bayesian network models.

Naive Bayes classifiers generally perform worse than more advanced models like logistic regressions, especially at quantifying uncertainty (with naive Bayes models often producing wildly overconfident probabilities). However, they are highly scalable, requiring only one parameter for each feature or predictor in a learning problem. Maximum-likelihood training can be done by evaluating a closed-form expression (simply by counting observations in each group),

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. There 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 10 cm 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.

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 real-world 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. 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.

An advantage of naive Bayes is that it only requires a small amount of training data to estimate the parameters necessary for classification.

Probabilistic model

Abstractly, naive Bayes is a conditional probability model: it assigns probabilities <math>p(C_k \mid x_1, \ldots, x_n)</math> for each of the possible outcomes or classes <math>C_k</math> given a problem instance to be classified, represented by a vector <math>\mathbf{x} = (x_1, \ldots, x_n)</math> encoding some features (independent variables).

The problem with the above formulation is that if the number of features is large or if a feature can take on a large number of values, then basing such a model on probability tables is infeasible. The model must therefore be reformulated to make it more tractable. Using Bayes' theorem, the conditional probability can be decomposed as:

<math display="block">p(C_k \mid \mathbf{x}) = \frac{p(C_k) \ p(\mathbf{x} \mid C_k)}{p(\mathbf{x})} \,</math>

In plain English, using Bayesian probability terminology, the above equation can be written as

<math display="block">\text{posterior} = \frac{\text{prior} \times \text{likelihood{\text{evidence \,</math>

In practice, there is interest only in the numerator of that fraction, because the denominator does not depend on <math>C</math> and the values of the features <math>x_i</math> are given, so that the denominator is effectively constant.

The numerator is equivalent to the joint probability model

<math display="block">p(C_k, x_1, \ldots, x_n)\,</math>

which can be rewritten as follows, using the chain rule for repeated applications of the definition of conditional probability:

<math display="block">\begin{align}

p(C_k, x_1, \ldots, x_n) & = p(x_1, \ldots, x_n, C_k) \\

& = p(x_1 \mid x_2, \ldots, x_n, C_k) \ p(x_2, \ldots, x_n, C_k) \\

& = p(x_1 \mid x_2, \ldots, x_n, C_k) \ p(x_2 \mid x_3, \ldots, x_n, C_k) \ p(x_3, \ldots, x_n, C_k) \\

& = \cdots \\

& = p(x_1 \mid x_2, \ldots, x_n, C_k) \ p(x_2 \mid x_3, \ldots, x_n, C_k) \cdots p(x_{n-1} \mid x_n, C_k) \ p(x_n \mid C_k) \ p(C_k) \\

\end{align}</math>

Now the "naive" conditional independence assumptions come into play: assume that all features in <math>\mathbf{x}</math> are mutually independent, conditional on the category <math>C_k</math>. Under this assumption,

<math display="block">p(x_i \mid x_{i+1}, \ldots ,x_{n}, C_k ) = p(x_i \mid C_k)\,.</math>

Thus, the joint model can be expressed as

<math display="block">\begin{align}

p(C_k \mid x_1, \ldots, x_n) \varpropto\ & p(C_k, x_1, \ldots, x_n) \\

& = p(C_k) \ p(x_1 \mid C_k) \ p(x_2\mid C_k) \ p(x_3\mid C_k) \ \cdots \\

& = p(C_k) \prod_{i=1}^n p(x_i \mid C_k)\,,

\end{align}</math>

where <math>\varpropto</math> denotes proportionality since the denominator <math>p(\mathbf{x})</math> is omitted.

This means that under the above independence assumptions, the conditional distribution over the class variable <math>C</math> is:

<math display="block">p(C_k \mid x_1, \ldots, x_n) = \frac{1}{Z} \ p(C_k) \prod_{i=1}^n p(x_i \mid C_k)</math>

where the evidence <math>Z = p(\mathbf{x}) = \sum_k p(C_k) \ p(\mathbf{x} \mid C_k)</math> is a scaling factor dependent only on <math>x_1, \ldots, x_n</math>, that is, a constant if the values of the feature variables are known.

Often, it is only necessary to discriminate between classes. In that case, the scaling factor is irrelevant, and it is sufficient to calculate the log-probability up to a factor:<math display="block">\ln p(C_k \mid x_1, \ldots, x_n) = \ln p(C_k) + \sum_{i=1}^n \ln p(x_i \mid C_k) \underbrace{- \ln Z}_{\text{irrelevant</math>The scaling factor is irrelevant, since discrimination subtracts it away:<math display="block">\ln \frac{p(C_k \mid x_1, \ldots, x_n)}{p(C_l \mid x_1, \ldots, x_n)} = \left(\ln p(C_k) + \sum_{i=1}^n \ln p(x_i \mid C_k) \right) - \left(\ln p(C_l) + \sum_{i=1}^n \ln p(x_i \mid C_l) \right)</math>There are two benefits of using log-probability. One is that it allows an interpretation in information theory, where log-probabilities are units of information in nats. Another is that it avoids arithmetic underflow.

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 so as to minimize the probability of misclassification; 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 label <math>\hat{y} = C_k</math> for some as follows:

<math display="block">\hat{y} = \underset{k \in \{1, \ldots, K\{\operatorname{argmax \ p(C_k) \displaystyle\prod_{i=1}^n p(x_i \mid C_k).</math>

thumb|[[Likelihood functions <math>p(\mathbf{x} \mid Y)</math>, Confusion matrix and ROC curve. For the naive Bayes classifier and given that the a priori probabilities <math>p(Y)</math> are the same for all classes, then the decision boundary (green line) would be placed on the point where the two probability densities intersect, due to ]]

Parameter estimation and event models

A class's prior may be calculated by assuming equiprobable classes, i.e., <math>p(C_k) = \frac{1}{K}</math>, or by calculating an estimate for the class probability from the training set:

<math display="block">\text{prior for a given class} = \frac{\text{no. of samples in that class{\text{total no. of samples \,</math>

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.

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.

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 normal (or Gaussian) distribution. For example, suppose the training data contains a continuous attribute, <math>x</math>. The data is first segmented by the class, and then the mean and variance of <math>x</math> is computed in each class. Let <math>\mu_k</math> be the mean of the values in <math>x</math> associated with class <math>C_k</math>, and let <math>\sigma^2_k</math> be the Bessel corrected variance of the values in <math>x</math> associated with class <math>C_k</math>. Suppose one has collected some observation value <math>v</math>. Then, the probability density of <math>v</math> given a class <math>C_k</math>, i.e., <math>p(x=v \mid C_k)</math>, can be computed by plugging <math>v</math> into the equation for a normal distribution parameterized by <math>\mu_k</math> and <math>\sigma^2_k</math>. Formally,

p(x=v \mid C_k) = \frac{1}{\sqrt{2\pi\sigma^2_k\,e^{ -\frac{(v-\mu_k)^2}{2\sigma^2_k} }

</math>

Another common technique for handling continuous values is to use binning to discretize the feature values and obtain a new set of Bernoulli-distributed features. Some literature suggests that this is required in order to use naive Bayes, but it is not true, as the discretization may throw away discriminative information.

Multinomial naive Bayes

With a multinomial event model, samples (feature vectors) represent the frequencies with which certain events have been generated by a multinomial <math>(p_1, \dots, p_n)</math> where <math>p_i</math> is the probability that event occurs (or such multinomials in the multiclass case). A feature vector <math>\mathbf{x} = (x_1, \dots, x_n)</math> is then a histogram, with <math>x_i</math> counting the number of times event 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 is given by:

p(\mathbf{x} \mid C_k) = \frac{(\sum_{i=1}^n x_i)!}{\prod_{i=1}^n x_i !} \prod_{i=1}^n {p_{ki^{x_i}

</math>

where <math>p_{ki} := p(i \mid C_k)</math>.

The multinomial naive Bayes classifier becomes a linear classifier when expressed in log-space:

\begin{align}

\log p(C_k \mid \mathbf{x}) & \varpropto \log \left( p(C_k) \prod_{i=1}^n {p_{ki^{x_i} \right) \\

& = \log p(C_k) + \sum_{i=1}^n x_i \cdot \log p_{ki} \\

& = b + \mathbf{w}_k^\top \mathbf{x}

\end{align}

</math>

where <math>b = \log p(C_k)</math> and <math>w_{ki} = \log p_{ki}</math>. Estimating the parameters in log space is advantageous since multiplying a large number of small values can lead to significant rounding error. Applying a log transform reduces the effect of this rounding error.

If a given class and feature value never occur together in the training data, then the frequency-based probability estimate will be zero, because the probability estimate is directly proportional to the number of occurrences of a feature's value. 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, 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.

Discussion

Despite the fact that the far-reaching 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 one-dimensional 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, 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 predicted as 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. Other reasons for the observed success of the naive Bayes classifier are discussed in the literature cited below.

Relation to logistic regression

In the case of discrete inputs (indicator or frequency features for discrete events), naive Bayes classifiers form a generative-discriminative pair with multinomial logistic regression classifiers: each naive Bayes classifier can be considered a way of fitting a probability model that optimizes the joint likelihood <math>p(C, \mathbf{x})</math>, while logistic regression fits the same probability model to optimize the conditional <math>p(C \mid \mathbf{x})</math>.

More formally, we have the following:

The link between the two can be seen by observing that the decision function for naive Bayes (in the binary case) can be rewritten as "predict class <math>C_1</math> if the odds of <math>p(C_1 \mid \mathbf{x})</math> exceed those of <math>p(C_2 \mid \mathbf{x})</math>". Expressing this in log-space gives:

\log\frac{p(C_1 \mid \mathbf{x})}{p(C_2 \mid \mathbf{x})} = \log p(C_1 \mid \mathbf{x}) - \log p(C_2 \mid \mathbf{x}) > 0

</math>

The left-hand side of this equation is the log-odds, or logit, the quantity predicted by the linear model that underlies logistic regression. Since naive Bayes is also a linear model for the two "discrete" event models, it can be reparametrised as a linear function <math>b + \mathbf{w}^\top x > 0</math>. Obtaining the probabilities is then a matter of applying the logistic function to <math>b + \mathbf{w}^\top x</math>, or in the multiclass case, the softmax function.

Discriminative classifiers have lower asymptotic error than generative ones; however, research by Ng and Jordan has shown that in some practical cases naive Bayes can outperform logistic regression because it reaches its asymptotic error faster. The first scholarly publication using the naive Bayes classifier for spam filtering was by Sahami et al. in 1998.

Variants of the basic technique have been implemented in a number of research works and commercial software products. Many modern mail clients implement Bayesian spam filtering. Users can also install separate email filtering programs. Server-side email filters, such as DSPAM, Rspamd, SpamAssassin, SpamBayes, Bogofilter, and ASSP, make use of Bayesian spam filtering techniques, and the functionality is sometimes embedded within mail server software itself. CRM114, often cited as a Bayesian filter, is not intended to use a Bayes filter in production, but includes the ″unigram″ feature for reference.

Dealing with rare words

In the case a word has never been met during the learning phase, both the numerator and the denominator are equal to zero, both in the general formula and in the spamicity formula. The software can decide to discard such words for which there is no information available.

More generally, the words that were encountered only a few times during the learning phase cause a problem, because it would be an error to trust blindly the information they provide. A simple solution is to simply avoid taking such unreliable words into account as well.

Applying again Bayes' theorem, and assuming the classification between spam and ham of the emails containing a given word ("replica") is a random variable with beta distribution, some programs decide to use a corrected probability:

:<math>\Pr'(S|W) = \frac{s \cdot \Pr(S) + n \cdot \Pr(S|W)}{s + n }</math>

where:

<math>\Pr'(S|W)</math> is the corrected probability for the message to be spam, knowing that it contains a given word ;
<math>s</math> is the strength we give to background information about incoming spam ;
<math>\Pr(S)</math> is the probability of any incoming message to be spam ;
<math>n</math> is the number of occurrences of this word during the learning phase ;
<math>\Pr(S|W)</math> is the spamicity of this word.

(Demonstration:)

This corrected probability is used instead of the spamicity in the combining formula.

This formula can be extended to the case where n is equal to zero (and where the spamicity is not defined), and evaluates in this case to <math>Pr(S)</math>.

Other heuristics

"Neutral" words like "the", "a", "some", or "is" (in English), or their equivalents in other languages, can be ignored. These are also known as Stop words. More generally, some bayesian filtering filters simply ignore all the words which have a spamicity next to 0.5, as they contribute little to a good decision. The words taken into consideration are those whose spamicity is next to 0.0 (distinctive signs of legitimate messages), or next to 1.0 (distinctive signs of spam). A method can be for example to keep only those ten words, in the examined message, which have the greatest absolute value |0.5 − pI|.

Some software products take into account the fact that a given word appears several times in the examined message, others don't.

Some software products use patterns (sequences of words) instead of isolated natural languages words. For example, with a "context window" of four words, they compute the spamicity of "Viagra is good for", instead of computing the spamicities of "Viagra", "is", "good", and "for". This method gives more sensitivity to context and eliminates the Bayesian noise better, at the expense of a bigger database.

Disadvantages

Depending on the implementation, Bayesian spam filtering may be susceptible to Bayesian poisoning, a technique used by spammers in an attempt to degrade the effectiveness of spam filters that rely on Bayesian filtering. A spammer practicing Bayesian poisoning will send out emails with large amounts of legitimate text (gathered from legitimate news or literary sources). Spammer tactics include insertion of random innocuous words that are not normally associated with spam, thereby decreasing the email's spam score, making it more likely to slip past a Bayesian spam filter. However, with (for example) Paul Graham's scheme only the most significant probabilities are used, so that padding the text out with non-spam-related words does not affect the detection probability significantly.

Words that normally appear in large quantities in spam may also be transformed by spammers. For example, «Viagra» would be replaced with «Viaagra» or «V!agra» in the spam message. The recipient of the message can still read the changed words, but each of these words is met more rarely by the Bayesian filter, which hinders its learning process. As a general rule, this spamming technique does not work very well, because the derived words end up recognized by the filter just like the normal ones.

Another technique used to try to defeat Bayesian spam filters is to replace text with pictures, either directly included or linked. The whole text of the message, or some part of it, is replaced with a picture where the same text is "drawn". The spam filter is usually unable to analyze this picture, which would contain the sensitive words like «Viagra». However, since many mail clients disable the display of linked pictures for security reasons, the spammer sending links to distant pictures might reach fewer targets. Also, a picture's size in bytes is bigger than the equivalent text's size, so the spammer needs more bandwidth to send messages directly including pictures. Some filters are more inclined to decide that a message is spam if it has mostly graphical contents. A solution used by Google in its Gmail email system is to perform an OCR (Optical Character Recognition) on every mid to large size image, analyzing the text inside.

References

External links

Book Chapter: Naive Bayes text classification, Introduction to Information Retrieval
Naive Bayes for Text Classification with Unbalanced Classes

Naive Bayes classifier

Introduction

Probabilistic model

Constructing a classifier from the probability model

Parameter estimation and event models

Gaussian naive Bayes

Multinomial naive Bayes

Discussion

Relation to logistic regression

Dealing with rare words

Other heuristics

Disadvantages

See also

References

Further reading

External links