Properties of Naive Bayes

where Bayes' rule (Equation 59, page 59 ) is applied in (122) and we drop the denominator in the last step because is the same for all classes and does not affect the argmax.

We can interpret
Equation 123 as a description of the
generative process we assume in Bayesian text
classification. To generate a document, we first choose
class with probability (top nodes in
and 13.5 ).
The two models
differ in the formalization of the second step, the
generation of the document given the class, corresponding to
the conditional distribution
:

where is the sequence of terms as it occurs in (minus terms that were excluded from the vocabulary) and is a binary vector of dimensionality that indicates for each term whether it occurs in or not.

It should now be clearer why we introduced the document space in Equation 112 when we defined the classification problem. A critical step in solving a text classification problem is to choose the document representation. and are two different document representations. In the first case, is the set of all term sequences (or, more precisely, sequences of term tokens). In the second case, is .

We cannot use
and 125 for text
classification directly.
For the Bernoulli model,
we would have to estimate
different
parameters, one for each possible combination of
values and a class. The number of parameters
in the
multinomial case has the same order of
magnitude.^{}This being a very large
quantity, estimating these parameters reliably is
infeasible.

To reduce the number of parameters,
we make
the Naive Bayes *conditional independence
assumption* . We assume that attribute values are independent of
each other given the class:

We have introduced two random variables here to make the two different generative models explicit.

We illustrate the conditional independence assumption in and 13.5 . The class China generates values for each of the five term attributes (multinomial) or six binary attributes (Bernoulli) with a certain probability, independent of the values of the other attributes. The fact that a document in the class China contains the term Taipei does not make it more likely or less likely that it also contains Beijing.

In reality, the conditional independence assumption does not
hold for text data. Terms *are* conditionally dependent
on each other. But as we will discuss shortly, NB models
perform well despite the conditional independence
assumption.

Even when assuming conditional independence, we still have too many parameters for the multinomial model if we assume a different probability distribution for each position in the document. The position of a term in a document by itself does not carry information about the class. Although there is a difference between China sues France and France sues China, the occurrence of China in position 1 versus position 3 of the document is not useful in NB classification because we look at each term separately. The conditional independence assumption commits us to this way of processing the evidence.

Also, if we assumed different term distributions for each position , we would have to estimate a different set of parameters for each . The probability of bean appearing as the first term of a coffee document could be different from it appearing as the second term, and so on. This again causes problems in estimation owing to data sparseness.

For these reasons, we make a second
independence assumption for the multinomial model,
*positional independence* :
The conditional probabilities for a term are the same
independent of position in the document.

(128) |

With conditional and positional independence assumptions, we only need to estimate parameters (multinomial model) or (Bernoulli model), one for each term-class combination, rather than a number that is at least exponential in , the size of the vocabulary. The independence assumptions reduce the number of parameters to be estimated by several orders of magnitude.

To summarize, we generate a document in the multinomial model (Figure 13.4 ) by first picking a class with where is a random variable taking values from as values. Next we generate term in position with for each of the positions of the document. The all have the same distribution over terms for a given . In the example in Figure 13.4 , we show the generation of , corresponding to the one-sentence document Beijing and Taipei join WTO.

For a completely specified document generation model, we would also have to define a distribution over lengths. Without it, the multinomial model is a token generation model rather than a document generation model.

We generate a document in the Bernoulli model (Figure 13.5 ) by first picking a class with and then generating a binary indicator for each term of the vocabulary ( ). In the example in Figure 13.5 , we show the generation of , corresponding, again, to the one-sentence document Beijing and Taipei join WTO where we have assumed that and is a stop word.

multinomial model | Bernoulli model | |||

event model | generation of token | generation of document | ||

random variable(s) | iff occurs at given pos | iff occurs in doc | ||

document representation | ||||

parameter estimation | ||||

decision rule: maximize | ||||

multiple occurrences | taken into account | ignored | ||

length of docs | can handle longer docs | works best for short docs | ||

# features | can handle more | works best with fewer | ||

estimate for term the |

We compare the two models in Table 13.3 , including estimation equations and decision rules.

Naive Bayes is so called because the
independence assumptions we have just made are indeed very
naive for a model of natural language. The conditional
independence assumption states that features are independent
of each other given the class. This is hardly ever true for
terms in documents. In many cases, the opposite is true.
The pairs hong and kong or london and
english in Figure 13.7 are examples of highly
dependent terms. In addition, the multinomial model makes an
assumption of positional independence. The Bernoulli model
ignores positions in documents altogether because it only
cares about absence or presence. This *bag-of-words*
model discards all information that is communicated by the
order of words in natural language sentences.
How can NB be a good text classifier when its model of natural
language is so oversimplified?

class selected | |||||

true probability | 0.6 | 0.4 | |||

(Equation 126) | 0.00099 | 0.00001 | |||

NB estimate | 0.99 | 0.01 |

The answer is that
even though the *probability estimates* of
NB are of low quality, its *classification
decisions* are surprisingly good. Consider a document
with true probabilities
and
as shown in Table 13.4 .
Assume that contains
many terms that are positive indicators for
and many terms that are negative indicators for .
Thus, when using the
multinomial model in Equation 126,
will be much larger than
(0.00099 vs. 0.00001 in the table).
After division by 0.001 to get well-formed probabilities
for , we end up with one estimate that is close to
1.0 and one that is close to 0.0. This is common:
The winning class in NB classification
usually has a much larger probability than the other
classes and the estimates diverge very significantly from
the true probabilities. But the
classification decision is based on which class gets the
highest score. It does not matter how accurate the
estimates are. Despite the bad estimates, NB
estimates a
higher probability for and therefore assigns
to the correct class in Table 13.4 . *Correct estimation implies
accurate prediction, but accurate prediction does not imply
correct estimation.* NB classifiers estimate badly,
but often classify well.

Even if it is not the method with the highest accuracy for
text, NB has many virtues that make it a strong
contender for text classification. It excels if there are
many equally important features that jointly contribute to
the classification decision. It is also somewhat robust to
noise features (as defined in the next
section) and *concept
drift* - the gradual change over
time of the concept underlying a class like US
president from Bill Clinton to George W. Bush (see
Section 13.7 ). Classifiers like
kNN
knn
can be carefully tuned to idiosyncratic properties of a
particular time period. This will then hurt them when
documents in the following time period have slightly
different properties.

The Bernoulli model is particularly robust with respect to concept drift. We will see in Figure 13.8 that it can have decent performance when using fewer than a dozen terms. The most important indicators for a class are less likely to change. Thus, a model that only relies on these features is more likely to maintain a certain level of accuracy in concept drift.

NB's main strength is its efficiency: Training and classification can be accomplished with one pass over the data. Because it combines efficiency with good accuracy it is often used as a baseline in text classification research. It is often the method of choice if (i) squeezing out a few extra percentage points of accuracy is not worth the trouble in a text classification application, (ii) a very large amount of training data is available and there is more to be gained from training on a lot of data than using a better classifier on a smaller training set, or (iii) if its robustness to concept drift can be exploited.

(1) | He moved from London, Ontario, to London, England. | ||

(2) | He moved from London, England, to London, Ontario. | ||

(3) | He moved from England to London, Ontario. |

In this book, we discuss NB as a classifier for
text. The independence assumptions do not hold for
text. However, it can be shown that NB is an
*optimal classifier*
(in the sense of minimal error rate on
new data) for data where the independence
assumptions do hold.

This is an automatically generated page. In case of formatting errors you may want to look at the PDF edition of the book.

2009-04-07