|
|
| Line 1: |
Line 1: |
| {{multiple issues|
| | == 'Big Brother ......' == |
| {{Cleanup|date=March 2013}}
| |
| {{External links|date=November 2011}}
| |
| }}
| |
|
| |
|
| {{Machine learning bar}}
| | If you promise to give me to eat, I'll put this man, a man can eat me no good at all, I do not bother to eat. Do クリスチャンルブタン メンズ you think my plan is good. In fact, I do not like クリスチャンルブタン 靴 メンズ the other barely Wicked. But ...... If you resist, it forced me to eat even the two of you together. 'Longyan lion's voice sounded in the Qin Yu and black to mind.<br><br>black lion Longyan feeling momentum, Jin Yan eagle than ten times the momentum of terror, knowing that simply can not match. Perhaps virtual hole of クリスチャンルブタン 取扱店 comprehension, the right of the クリスチャンルブタン バッグ animal on the Yuan Ying lion Longyan not necessarily have to grasp, let alone the Qin Yu and black.<br><br>black look to the Qin Yu, itchy eyes exudes the emotional complexity of people and land, Qin Yu was very clearly feel.<br><br>'Big Brother ......'<br><br>'black, Freeze want.' Qin クリスチャンルブタン 銀座 Yu Lenghe voice sounded in the black mind, 'Black, if you do not spell it, so for me to let you die, I will be sad to live a lifetime of guilt. What's more ...... we are not without hope of escape! '<br><br>Qin Yu heart |
| A '''naive Bayes classifier''' is a simple probabilistic [[Statistical classification|classifier]] based on applying [[Bayes' theorem]] with strong (naive) [[statistical independence|independence]] assumptions. A more descriptive term for the underlying probability model would be "[[statistical independence|independent]] feature model". An overview of statistical classifiers is given in the article on [[Pattern recognition]].
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>http://tonzawa.godream.ne.jp/cgi-bin/aska/aska.cgi</li> |
| | |
| | <li> |
| | |
| | <li> |
| | |
| | </ul> |
|
| |
|
| ==Introduction== | | == friendship == |
| In simple terms, a naive Bayes classifier assumes that the presence or absence of a particular feature is unrelated to the presence or absence 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 the presence or absence of the other 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.
| | 'Destruction of any of the relief channel in relief, is a capital offense. 'This one message, Qin Yu is dumbfounding. Although last a relief, let Qin Yu heart shock.<br><br>But the whole relief channels relief too much, especially the beginning of the relief, one on skills, are able to estimate クリスチャンルブタン 値段 any fairy higher than the many. Then the 'bad' relief, the king a god of destruction is a capital offense. God King's life, in the eyes of who formulate rules really are not worth the money.<br><br>talking about them, God King who has consciously thirty-two combine to form a small group. The battle is about to begin competing, they can expect there will be more クリスチャンルブタン バッグ fierce, which is divided into three in order to revere Lingbao, probably the vast majority of God, the king will be ruthless killings.<br><br>friendship?<br><br>face?<br><br>the opportunity before becoming クリスチャンルブタン 価格 Senior, is a joke. No one will take into account the face, we fight クリスチャンルブタン アウトレット all life. Even if they know the success of only one. However, in order that a quota, we クリスチャンルブタン 取扱店 are willing to strive and work hard.<br><br>'thundered ......' |
| | 相关的主题文章： |
| | <ul> |
| | |
| | <li>http://home.kanto-gakuin.ac.jp/~kg068601/cgi-bin/i_light.cgi</li> |
| | |
| | <li>?aid=133</li> |
| | |
| | <li>?mod=viewthread&tid=194186&extra=</li> |
| | |
| | </ul> |
|
| |
|
| 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.<ref>{{cite conference | first = Harry | last = Zhang | title = The Optimality of Naive Bayes | conference = FLAIRS2004 conference | url = http://www.cs.unb.ca/profs/hzhang/publications/FLAIRS04ZhangH.pdf }}</ref> 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]].<ref>{{cite conference | last1 = Caruana | first1 = R. | last2 = Niculescu-Mizil | first2 = A. | title = An empirical comparison of supervised learning algorithms | booktitle = Proceedings of the 23rd international conference on Machine learning | year = 2006 | id = {{citeseerx|10.1.1.122.5901}} }}</ref>
| | == ' Ao dry and flow diagrams to see the magic Huang Peng == |
|
| |
|
| An advantage of Naive Bayes is that it only requires a small amount of training data to estimate the parameters (means and variances of the variables) necessary for classification. Because independent variables are assumed, only the variances of the variables for each class need to be determined and not the entire [[covariance matrix]].
| | Do not.<br><br>'Your Majesty?' Ao dry and flow diagrams to [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_3.php クリスチャンルブタン 東京] see the magic Huang Peng, Peng decided to wait for the magic Emperor.<br><br>magic Huang Peng began rising arrogance of so many years, and in front of these people could be so ignored him, shook his fists Wong Peng magic, golden body began to shine, but ......<br><br>little red dot in Scarlet magic Wong Peng surface float on top of the head Wong Peng magic golden crown had suddenly turned into a blood red crown.<br><br>'On shenfa sophisticated, yet few people dare to take me more than that you and I close of the war, [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_9.php クリスチャンルブタン] I do not believe that you can touch me.' magic Huang Peng blood red eyes and a faint glow.<br><br>house blue face changed, his mouth muttering: 'This is Peng family really is enough to seniors under the capital's ah, this takes a great effort.'<br><br>see his face solemn blue house, Wong Peng magic but relaxed mind, he understands [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_12.php クリスチャンルブタン メンズ] people do not deal with their immediate grasp. Magic Wong Peng immediately ordered: 'Xuan Xi, [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_2.php クリスチャンルブタン 日本] Ao dry, flow diagrams you three Weisha Qin Yu, I deal with this guy.'<br><br>'是.' Ao dry [http://www.lamartcorp.com/modules/mod_menu/rakuten_cl_9.php クリスチャンルブタン セール] and flow diagram should |
| | | 相关的主题文章： |
| == Probabilistic model ==
| | <ul> |
| | | |
| Abstractly, the probability model for a classifier is a conditional model.
| | <li>[http://www.i72.me/plus/feedback.php?aid=844 http://www.i72.me/plus/feedback.php?aid=844]</li> |
| | | |
| :<math>p(C \vert F_1,\dots,F_n)\,</math>
| | <li>[http://qdjieheng.com.cn/plus/feedback.php?aid=14 http://qdjieheng.com.cn/plus/feedback.php?aid=14]</li> |
| | | |
| 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.
| | <li>[http://www.wfyongdong.com/plus/feedback.php?aid=115 http://www.wfyongdong.com/plus/feedback.php?aid=115]</li> |
| | | |
| Using [[Bayes' theorem]], this can be written
| | </ul> |
| | |
| :<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 plain English the above equation can be written as
| |
| :<math>\mbox{posterior} = \frac{\mbox{prior} \times \mbox{likelihood}}{\mbox{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>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 the [[Chain rule (probability)|chain rule]] for 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>
| |
| | |
| ::<math>= p(C) \ p(F_1\vert C) \ p(F_2\vert C, F_1) \ \dots p(F_n\vert C, F_1, F_2, F_3,\dots,F_{n-1}).</math>
| |
| | |
| Now the "naive" [[conditional independence]] assumptions come into play: assume that each feature <math>F_i</math> is conditionally [[statistical independence|independent]] of every other feature <math>F_j</math> for <math>j\neq i</math> given the category <math>C</math>. This means that
| |
| | |
| :<math>p(F_i \vert C, F_j) = p(F_i \vert C)\,</math>, <math>p(F_i \vert C, F_j,F_k) = p(F_i \vert C)\,</math> , <math>p(F_i \vert C, F_j,F_k,F_l) = p(F_i \vert C)\,</math>, and so on,
| |
| | |
| for <math>i\ne j,k,l</math>, and so the joint model can be expressed as | |
| | |
| :<math> \begin{align}
| |
| p(C \vert F_1, \dots, F_n)
| |
| & \varpropto p(C, F_1, \dots, F_n) \\
| |
| & \varpropto p(C) \ p(F_1\vert C) \ p(F_2\vert C) \ p(F_3\vert C) \ \cdots \\
| |
| & \varpropto p(C) \prod_{i=1}^n p(F_i \vert C)\,.
| |
| \end{align}</math>
| |
| | |
| This means that under the above independence assumptions, the conditional distribution over the class variable <math>C</math> is:
| |
| | |
| :<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> (the evidence) is a scaling factor dependent only on <math>F_1,\dots,F_n</math>, that is, a constant if the values of the feature variables are known.
| |
| | |
| Models of this form are much more manageable, because 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 each <math>p(F_i\vert C=c)</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 distribution|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 classification.
| |
| | |
| == Parameter estimation and event models ==
| |
| | |
| All model parameters (''i.e.'', class priors and feature probability distributions) can be approximated with relative frequencies from the training set. These are [[maximum likelihood]] estimates of the probabilities. 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.<ref>George H. John and Pat Langley (1995). Estimating Continuous Distributions in Bayesian Classifiers. Proceedings of the Eleventh Conference on Uncertainty in Artificial Intelligence. pp. 338-345. Morgan Kaufmann, San Mateo.</ref>
| |
| | |
| 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 distribution|multinomial]] and [[Bernoulli distribution|Bernoulli]] distributions are popular. These assumptions lead to two distinct models, which are often confused.<ref>McCallum, Andrew, and Kamal Nigam. "A comparison of event models for Naive Bayes text classification." AAAI-98 workshop on learning for text categorization. Vol. 752. 1998.</ref><ref>Metsis, Vangelis, Ion Androutsopoulos, and Georgios Paliouras. "Spam filtering with Naive Bayes—which Naive Bayes?" Third conference on email and anti-spam (CEAS). Vol. 17. 2006.</ref>
| |
| When dealing with continuous data, a typical assumption is that the continuous values associated with each class are distributed according to a [[Normal distribution|Gaussian]] distribution.
| |
| | |
| For example, suppose the training data contain a continuous attribute, <math>x</math>. We first segment the data by the class, and then compute the mean and [[Variance#Estimating_the_variance|variance]] of <math>x</math> in each class. Let <math>\mu_c</math> be the mean of the values in <math>x</math> associated with class ''c'', and let <math>\sigma^2_c</math> be the variance of the values in <math>x</math> associated with class ''c''. Then, the probability ''density'' of some value given a class, <math>P(x=v|c)</math>, can be computed by plugging <math>v</math> into the equation for a [[Normal distribution]] parameterized by <math>\mu_c</math> and <math>\sigma^2_c</math>. That is,
| |
| | |
| <math> | |
| P(x=v|c)=\tfrac{1}{\sqrt{2\pi\sigma^2_c}}\,e^{ -\frac{(v-\mu_c)^2}{2\sigma^2_c} }
| |
| </math> | |
| | |
| Another common technique for handling continuous values is to use binning to [[Discretization of continuous features|discretize]] the feature values, to obtain a new set of Bernoulli-distributed features. In general, the distribution method is a better choice if there is a small amount of training data, or if the precise distribution of the data is known. The discretization method tends to do better if there is a large amount of training data because it will learn to fit the distribution of the data. Since naive Bayes is typically used when a large amount of data is available (as more computationally expensive models can generally achieve better accuracy), the discretization method is generally preferred over the distribution method.
| |
| | |
| == Sample correction ==
| |
| | |
| If a given class and feature value never occur together in the training data, then the frequency-based 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 small-sample correction, called [[pseudocount]], in all probability estimates such that no probability is ever set to be exactly zero.
| |
| | |
| == 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 [[Statistical classification|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 <math>\mathrm{classify}</math> defined as follows:
| |
| | |
| :<math>\mathrm{classify}(f_1,\dots,f_n) = \underset{c}{\operatorname{argmax}} \ p(C=c) \displaystyle\prod_{i=1}^n p(F_i=f_i\vert C=c).</math>
| |
| | |
| == 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.<ref>[http://www.egmont-petersen.nl/classifiers.htm An introductory tutorial to classifiers (introducing the basic terms, with numeric example)]</ref> 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 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.
| |
| | |
| == Examples ==
| |
| | |
| ===Sex classification===
| |
| Problem: classify whether a given person is a male or a female based on the measured features.
| |
| The features include height, weight, and foot size.
| |
| | |
| ====Training====
| |
| Example training set below.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! sex !! height (feet) !! weight (lbs) !! foot size(inches)
| |
| |-
| |
| | male || 6 || 180 || 12
| |
| |-
| |
| | male || 5.92 (5'11") || 190 || 11
| |
| |-
| |
| | male || 5.58 (5'7") || 170 || 12
| |
| |-
| |
| | male || 5.92 (5'11") || 165 || 10
| |
| |-
| |
| | female || 5 || 100 || 6
| |
| |-
| |
| | female || 5.5 (5'6") || 150 || 8
| |
| |-
| |
| | female || 5.42 (5'5") || 130 || 7
| |
| |-
| |
| | female || 5.75 (5'9") || 150 || 9
| |
| |}
| |
| | |
| The classifier created from the training set using a Gaussian distribution assumption would be (given variances are [[Variance#Population variance and sample variance|sample variances]]):
| |
| {| class="wikitable"
| |
| |-
| |
| ! sex !! mean (height) !! variance (height) !! mean (weight) !! variance (weight) !! mean (foot size) !! variance (foot size)
| |
| |-
| |
| | male || 5.855 || 3.5033e-02 || 176.25 || 1.2292e+02 || 11.25 || 9.1667e-01
| |
| |-
| |
| | female || 5.4175 || 9.7225e-02 || 132.5 || 5.5833e+02 || 7.5 || 1.6667e+00
| |
| |}
| |
| | |
| Let's say we have equiprobable classes so P(male)= P(female) = 0.5. This prior probability distribution might be based on our knowledge of frequencies in the larger population, or on frequency in the training set.
| |
| | |
| ====Testing====
| |
| | |
| Below is a sample to be classified as a male or female.
| |
| | |
| {| class="wikitable"
| |
| |-
| |
| ! sex !! height (feet) !! weight (lbs) !! foot size(inches)
| |
| |-
| |
| | sample || 6 || 130 || 8
| |
| |}
| |
| | |
| We wish to determine which posterior is greater, male or female. For the classification as male the posterior is given by
| |
| | |
| :<math>
| |
| posterior (male) = \frac{P(male) \, p(height | male) \, p(weight | male) \, p(foot size | male)}{evidence}
| |
| </math>
| |
| | |
| For the classification as female the posterior is given by
| |
| :<math>
| |
| posterior (female) = \frac{P(female) \, p(height | female) \, p(weight | female) \, p(foot size | female)}{evidence}
| |
| </math>
| |
| | |
| The evidence (also termed normalizing constant) may be calculated:
| |
| | |
| :<math>
| |
| evidence = P(male) \, p(height | male) \, p(weight | male) \, p(foot size | male) </math>
| |
| :<math>
| |
| + P(female) \, p(height | female) \, p(weight | female) \, p(foot size | female)
| |
| </math>
| |
| | |
| However, given the sample the evidence is a constant and thus scales both posteriors equally. It therefore does not affect classification and can be ignored. We now determine the probability distribution for the sex of the sample.
| |
| | |
| :<math>
| |
| P(male) = 0.5
| |
| </math>
| |
| :<math>p(\mbox{height} | \mbox{male}) = \frac{1}{\sqrt{2\pi \sigma^2}}\exp\left(\frac{-(6-\mu)^2}{2\sigma^2}\right) \approx 1.5789</math>,
| |
| | |
| where <math>\mu = 5.855</math> and <math>\sigma^2 = 3.5033e-02</math> are the parameters of normal distribution which have been previously determined from the training set. Note that a value greater than 1 is OK here – it is a probability density rather than a probability, because height is a continuous variable.
| |
| | |
| :<math>
| |
| p(\mbox{weight} | \mbox{male}) = 5.9881e-06
| |
| </math>
| |
| :<math>
| |
| p(\mbox{foot size} | \mbox{male}) = 1.3112e-3
| |
| </math>
| |
| :<math>
| |
| \mbox{posterior numerator (male)} = \mbox{their product} = 6.1984e-09
| |
| </math>
| |
| | |
| :<math>
| |
| P(\mbox{female}) = 0.5
| |
| </math> | |
| :<math>
| |
| p(\mbox{height} | \mbox{female}) = 2.2346e-1
| |
| </math> | |
| :<math>
| |
| p(\mbox{weight} | \mbox{female}) = 1.6789e-2
| |
| </math>
| |
| :<math>
| |
| p(\mbox{foot size} | \mbox{female}) = 2.8669e-1
| |
| </math>
| |
| :<math>
| |
| \mbox{posterior numerator (female)} = \mbox{their product} = 5.3778e-04
| |
| </math>
| |
| | |
| Since posterior numerator is greater in the female case, we predict the sample is female.
| |
| | |
| ===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 [[spamming|spam]] and non-spam [[e-mail]]s. 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 words are randomly distributed in the document - that is, words are not dependent on the length of the document, position within the document with relation to other words, or other document-context.)
| |
| | |
| Then the probability that a given document ''D'' contains all of the words <math>w_i</math>, 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''?" In other words, what is <math>p(C \vert D)\,</math>?
| |
| | |
| Now [[Conditional probability|by definition]]
| |
| | |
| :<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 mutually exclusive classes, ''S'' and ¬''S'' (e.g. spam and not spam), such that every element (email) is in either one or the other;
| |
| | |
| :<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\vert D)\over p(\neg S\vert D)}={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 function|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 ratio]]s" 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.)
| |
| | |
| Finally, the document can be classified as follows. It is spam if <math>p(S\vert D) > p(\neg S\vert D)</math> (i.e., <math>\ln{p(S\vert D)\over p(\neg S\vert D)} > 0</math>), otherwise it is not spam.
| |
| | |
| == See also ==
| |
| * [[AODE]]
| |
| * [[Bayesian spam filtering]]
| |
| * [[Bayesian network]]
| |
| * [[Random naive Bayes]]
| |
| * [[Linear classifier]]
| |
| * [[Bayesian inference]] (esp. as Bayesian techniques relate to [[Spam (e-mail)|spam]])
| |
| * [[Boosting (meta-algorithm)]]
| |
| * [[Fuzzy logic]]
| |
| * [[Logistic regression]]
| |
| * [[Class membership probabilities]]
| |
| * [[Neural network]]s
| |
| * [[Predictive analytics]]
| |
| * [[Perceptron]]
| |
| * [[Support vector machine]]
| |
| * [[Feature selection]]
| |
| | |
| ==References==
| |
| {{More footnotes|date=May 2009}}
| |
| {{reflist}}
| |
| | |
| ===Further reading===
| |
| * Domingos, Pedro & Michael Pazzani (1997) "On the optimality of the simple Bayesian classifier under zero-one loss". ''Machine Learning'', 29:103–137. ''(also online at [http://citeseer.ist.psu.edu/ CiteSeer]: [http://citeseer.ist.psu.edu/domingos97optimality.html])''
| |
| * Rish, Irina. (2001). "An empirical study of the naive Bayes classifier". IJCAI 2001 Workshop on Empirical Methods in Artificial Intelligence. ''(available online: [http://www.research.ibm.com/people/r/rish/papers/RC22230.pdf PDF], [http://www.research.ibm.com/people/r/rish/papers/ijcai-ws.ps PostScript])''
| |
| * Hand, DJ, & Yu, K. (2001). "Idiot's Bayes - not so stupid after all?" International Statistical Review. Vol 69 part 3, pages 385-399. ISSN 0306-7734.
| |
| * Webb, G. I., J. Boughton, and Z. Wang (2005). [http://www.springerlink.com/content/u8w306673m1p866k/ Not So Naive Bayes: Aggregating One-Dependence Estimators]. Machine Learning 58(1). Netherlands: Springer, pages 5–24.
| |
| * Mozina M, Demsar J, Kattan M, & Zupan B. (2004). "Nomograms for Visualization of Naive Bayesian Classifier". In Proc. of PKDD-2004, pages 337-348. ''(available online: [http://www.ailab.si/blaz/papers/2004-PKDD.pdf PDF])''
| |
| * Maron, M. E. (1961). "Automatic Indexing: An Experimental Inquiry." Journal of the ACM (JACM) 8(3):404–417. ''(available online: [http://delivery.acm.org/10.1145/330000/321084/p404-maron.pdf?key1=321084&key2=9636178211&coll=GUIDE&dl=ACM&CFID=56729577&CFTOKEN=37855803 PDF])''
| |
| * Minsky, M. (1961). "Steps toward Artificial Intelligence." Proceedings of the IRE 49(1):8-30.
| |
| * McCallum, A. and Nigam K. "A Comparison of Event Models for Naive Bayes Text Classification". In AAAI/ICML-98 Workshop on Learning for Text Categorization, pp. 41–48. Technical Report WS-98-05. AAAI Press. 1998. ''(available online: [http://www.kamalnigam.com/papers/multinomial-aaaiws98.pdf PDF])''
| |
| * Rennie J, Shih L, Teevan J, and Karger D. Tackling The Poor Assumptions of Naive Bayes Classifiers. In Proceedings of the Twentieth International Conference on Machine Learning (ICML). 2003. ''(available online: [http://people.csail.mit.edu/~jrennie/papers/icml03-nb.pdf PDF])''
| |
| | |
| ==External links== | |
| * [http://nlp.stanford.edu/IR-book/html/htmledition/naive-bayes-text-classification-1.html Book Chapter: Naive Bayes text classification, Introduction to Information Retrieval]
| |
| * [http://www.cs.waikato.ac.nz/~eibe/pubs/FrankAndBouckaertPKDD06new.pdf Naive Bayes for Text Classification with Unbalanced Classes]
| |
| * [http://tunedit.org/results?d=UCI/&a=bayes Benchmark results of Naive Bayes implementations]
| |
| * [http://www.biomedcentral.com/1471-2105/7/514 Hierarchical Naive Bayes Classifiers for uncertain data] (an extension of the Naive Bayes classifier).
| |
| * [http://www.lwebzem.com/cgi-bin/res/naive_bayes_tm_classifier.cgi Document Classification Using Naive Bayes Classifier with Perl]
| |
| ;Software
| |
| * Naive Bayes classifiers are available in many general-purpose machine learning and NLP packages, including [[Apache Mahout]], [http://mallet.cs.umass.edu/ Mallet], [[NLTK]], [[Orange (software)|Orange]], [[scikit-learn]] and [[Weka (machine learning)|Weka]].
| |
| * [[IMSL Numerical Libraries]] Collections of math and statistical algorithms available in C/C++, Fortran, Java and C#/.NET. Data mining routines in the IMSL Libraries include a Naive Bayes classifier.
| |
| * [http://doc.winnowtag.org/open-source Winnow content recommendation] Open source Naive Bayes text classifier works with very small training and unbalanced training sets. High performance, C, any Unix.
| |
| * An interactive [[Microsoft Excel]] spreadsheet [http://downloads.sourceforge.net/naivebayesclass/NaiveBayesDemo.xls?use_mirror=osdn Naive Bayes implementation] using [[Visual Basic for Applications|VBA]] (requires enabled macros) with viewable source code.
| |
| * [http://jbnc.sourceforge.net/ jBNC - Bayesian Network Classifier Toolbox]
| |
| * [http://cmp.felk.cvut.cz/cmp/software/stprtool/ Statistical Pattern Recognition Toolbox for Matlab].
| |
| * [http://people.csail.mit.edu/jrennie/ifile/ ifile] - the first freely available (Naive) Bayesian mail/spam filter
| |
| * [http://nclassifier.sourceforge.net/ NClassifier] - NClassifier is a .NET library that supports text classification and text summarization. It is a port of Classifier4J.
| |
| * [http://classifier4j.sourceforge.net/ Classifier4J] - Classifier4J is a Java library designed to do text classification. It comes with an implementation of a Bayesian classifier.
| |
| | |
| [[Category:Classification algorithms]]
| |
| [[Category:Bayesian statistics]]
| |
| [[Category:Statistical classification]]
| |