Naive Bayesian classification

Naive Bayesian classification is the popular name for a probabilistic classification method. A more descriptive term for the underlying probability model is independent feature model. The term Naive Bayes refers to the fact that the probability model can be derived using Bayes' Theorem (credited to Thomas Bayes) and that it incorporates strong independences assumption that often have no bearing in reality, hence are (deliberately) naive. 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.

Abstractly, the desired 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 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

In practice we are only interested in the numerator of that fraction, since 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 the joint probability model

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

and so forth. Now the "naive" conditional independence assumptions come into play: assume that each feature is conditionally independent of every other feature (). This means that

and so the joint model can be expressed as

This means that under the above independence assumptions, the conditional distribution over the class variable can be expressed like this:

where is a scaling factor dependent only on , i.e., a constant if the values of the feature variables are known.

Models of this form tend to be much more manageable, since they factor into a so-called class prior and independent probability distributions . If there are classes and if a model for can be expressed in terms of parameters, then the corresponding Naive Bayes model has parameters. In practice, often (binary classification) and (Bernoulli variables as features) are common, and so the total number of parameters of the Naive Bayes model is , where is the number of binary features used for prediction.

The discussion so far has derived the independent feature model a.k.a. 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 defined as follows:

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.

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.

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

(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

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)

and

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

Assume for the moment that there are only two classes, S and ¬S.

and

Using the Bayesian result above, we can write:

Dividing one by the other gives:

Which can be re-factored as:

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 ln(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:

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.

In real life, the naive Bayes approach is more powerful than might be expected from the extreme simplicity of its model; in particular, it is fairly robust in the presence of non-independent attributes w_i. Recent theoretical analysis has shown why the naive Bayes classifier is so robust.

External links

Naive Bayesian learning paper
http://www.cs.washington.edu/ai/naive.html
An empirical study of the naive Bayes classifier (in pdf format)
On the Optimality of the Simple Bayesian Classifier under Zero-One Loss