Naive Bayes classifier
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A naive Bayes classifier (also known as Idiot's Bayes) is a simple probabilistic classifier. Naive Bayes classifiers are based on probability models that incorporate strong independence assumptions which often have no bearing in reality, hence are (deliberately) naive. A more descriptive term for the underlying probability model would be independent feature model. Furthermore the probability model can be derived using Bayes' Theorem (credited to Thomas Bayes).
Depending on the precise nature of the probability model, 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 believing in Bayesian probability or using any Bayesian methods.
In spite of their naive design and apparently over-simplified assumptions, naive Bayes classifiers often work much better in many complex real-world situations than might be expected from their very simple design. Recently, careful analysis of the Bayesian classification problem has shown that there are sound theoretical reasons for the apparently unreasonable efficacy of naive Bayes classifiers (see references at the end of this article for more details).
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The naive Bayes probabilistic model
Abstractly, the probability model for a classifier is a conditional model
- <math>p(C \vert F_1,\dots,F_n)\,<math>
over a dependent class variable <math>C<math> with a small number of outcomes or classes, conditional on several feature variables <math>F_1<math> through <math>F_n<math>. The problem is that if the number of features <math>n<math> is large or when 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, we write
- <math>p(C \vert F_1,\dots,F_n) = \frac{p(C) \ p(F_1,\dots,F_n\vert C)}{p(F_1,\dots,F_n)}. \,<math>
In practice we are only interested in the numerator of that fraction, since the denominator does not depend on <math>C<math> and the values of the features <math>F_i<math> are given, so that the denominator is effectively constant. The numerator is equivalent to the joint probability model
- <math>p(C, F_1, \dots, F_n)\,<math>
which can be rewritten as follows, using repeated applications of the definition of conditional probability:
- <math>p(C, F_1, \dots, F_n)\,<math>
- <math>= p(C) \ p(F_1,\dots,F_n\vert C)<math>
- <math>= p(C) \ p(F_1\vert C) \ p(F_2,\dots,F_n\vert C, F_1)<math>
- <math>= p(C) \ p(F_1\vert C) \ p(F_2\vert C, F_1) \ p(F_3,\dots,F_n\vert C, F_1, F_2)<math>
- <math>= p(C) \ p(F_1\vert C) \ p(F_2\vert C, F_1) \ p(F_3\vert C, F_1, F_2) \ p(F_4,\dots,F_n\vert C, F_1, F_2, F_3)<math>
and so forth. Now the "naive" conditional independence assumptions come into play: assume that each feature <math>F_i<math> is conditionally independent of every other feature <math>F_j<math> for <math>j\neq i<math>. This means that
- <math>p(F_i \vert C, F_j) = p(F_i \vert C)\,<math>
and so the joint model can be expressed as
- <math>p(C, F_1, \dots, F_n)
= p(C) \ p(F_1\vert C) \ p(F_2\vert C) \ p(F_3\vert C) \ \cdots\,<math>
- <math>= p(C) \prod_{i=1}^n p(F_i \vert C).\,<math>
This means that under the above independence assumptions, the conditional distribution over the class variable <math>C<math> can be expressed like this:
- <math>p(C \vert F_1,\dots,F_n) = \frac{1}{Z} p(C) \prod_{i=1}^n p(F_i \vert C)<math>
where <math>Z<math> is a scaling factor dependent only on <math>F_1,\dots,F_n<math>, i.e., a constant if the values of the feature variables are known.
Models of this form are much more manageable, since they factor into a so-called class prior <math>p(C)<math> and independent probability distributions <math>p(F_i\vert C)<math>. If there are <math>k<math> classes and if a model for <math>p(F_i)<math> can be expressed in terms of <math>r<math> parameters, then the corresponding naive Bayes model has (k − 1) + n r k parameters. In practice, often <math>k=2<math> (binary classification) and <math>r=1<math> (Bernoulli variables as features) are common, and so the total number of parameters of the naive Bayes model is <math>2n+1<math>, where <math>n<math> is the number of binary features used for prediction.
Parameter estimation
In a supervised learning setting, one wants to estimate the parameters of the probability model. Because of the independent feature assumption, it suffices to estimate the class prior and the conditional feature models independently, using the method of maximum likelihood, Bayesian inference or other parameter estimation procedures.
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 is the function <math>\mathrm{classify}<math> defined as follows:
- <math>\mathrm{classify}(f_1,\dots,f_n) = \mathop{\mathrm{argmax}}_c \ p(C=c) \prod_{i=1}^n p(F_i=f_i\vert C=c)<math>
Discussion
The naive Bayes classifier has several properties that make it surprisingly useful in practice, despite the fact that the far-reaching independence assumptions are often violated. Like all probabilistic classifiers under the MAP decision rule, it arrives at the correct classification as long as the correct class is more probable than any other class; class probabilities do not have to be estimated very well. In other words, the overall classifier is robust to serious deficiencies of its underlying naive probability model. Other reasons for the observed success of the naive Bayes classifier are discussed in the literature cited below.
Example: document classification
Here is a worked example of naive Bayesian classification to the document classification problem. Consider the problem of classifying documents by their content, for example into spam and non-spam E-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 C can be written as
- <math>p(w_i \vert C)\,<math>
(For this treatment, we simplify things further by assuming that the probability of a word in a document is independent of the length of a document, or that all documents are of the same length).
Then the probability of a given document D, given a class C, is
- <math>p(D\vert C)=\prod_i p(w_i \vert C)\,<math>
The question that we desire to answer is: "what is the probability that a given document D belongs to a given class C?"
Now, by their definition, (see Probability axiom)
- <math>p(D\vert C)={p(D\cap C)\over p(C)}<math>
and
- <math>p(C\vert D)={p(D\cap C)\over p(D)}<math>
Bayes' theorem manipulates these into a statement of probability in terms of likelihood.
- <math>p(C\vert D)={p(C)\over p(D)}\,p(D\vert C)<math>
Assume for the moment that there are only two classes, S and ¬S.
- <math>p(D\vert S)=\prod_i p(w_i \vert S)\,<math>
and
- <math>p(D\vert\neg S)=\prod_i p(w_i\vert\neg S)\,<math>
Using the Bayesian result above, we can write:
- <math>p(S\vert D)={p(S)\over p(D)}\,\prod_i p(w_i \vert S)<math>
- <math>p(\neg S\vert D)={p(\neg S)\over p(D)}\,\prod_i p(w_i \vert\neg S)<math>
Dividing one by the other gives:
- <math>{p(S\vert D)\over p(\neg S\vert D)}={p(S)\,\prod_i p(w_i \vert S)\over p(\neg S)\,\prod_i p(w_i \vert\neg S)}<math>
Which can be re-factored as:
- <math>{p(S)\over p(\neg S)}\,\prod_i {p(w_i \vert S)\over p(w_i \vert\neg S)}<math>
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 the logarithm of all these ratios, we have:
- <math>\ln{p(S\vert D)\over p(\neg S\vert D)}=\ln{p(S)\over p(\neg S)}+\sum_i \ln{p(w_i\vert S)\over p(w_i\vert\neg S)}<math>
This technique of "log-likelihood ratios" is a common technique in statistics. In the case of two mutually exclusive alternatives (such as this example), the conversion of a log-likelihood ratio to a probability takes the form of a sigmoid curve: see logit for details.
See also
- Linear classifier
- Bayesian inference (esp. as Bayesian techniques relate to spam)
- Boosting
- Fuzzy logic
- Logistic regression
- Neural networks
- Perceptron
- Support vector machine
References
- Pedro Domingos and Michael Pazzani. "On the optimality of the simple Bayesian classifier under zero-one loss". Machine Learning, 29:103-130, 1997. (also online at CiteSeer (http://citeseer.nj.nec.com): [1] (http://citeseer.nj.nec.com/domingos97optimality.html))
- Irina Rish. "An empirical study of the naive Bayes classifier". IJCAI 2001 Workshop on Empirical Methods in Artificial Intelligence. (available online: PDF (http://www.intellektik.informatik.tu-darmstadt.de/~tom/IJCAI01/Rish.pdf), PostScript (http://www.research.ibm.com/people/r/rish/papers/ijcai-ws.ps))
- Hand, DJ, and Yu, K. "Idiot's Bayes - not so stupid after all?" International Statistical Review. 2001. Vol 69 part 3, pages 385-399. ISSN 0306 7734.
- Mozina M, Demsar J, Kattan M, Zupan B. "Nomograms for Visualization of Naive Bayesian Classifier". In Proc. of PKDD-2004, pages 337-348. (available online: PDF (http://www.ailab.si/blaz/papers/2004-PKDD.pdf))
External links
- Naive Bayesian learning (http://citeseer.nj.nec.com/30545.html)