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FIFA World Rankings
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This article is about the men's rankings. For the women's rankings, see FIFA Women's World Rankings.
Top 20 rankings as of 7 June 2018[1]
Rank
Change
Team
Points
1 Germany 1558
2 Brazil 1431
3 Belgium 1298
4 Portugal 1274
5 Argentina 1241
6 Switzerland 1199
7 France 1198
8 2 Poland 1183
9 Chile 1135
10 2 Spain 1126
11 Peru 1125
12 Denmark 1051
12 1 England 1051
14 3 Uruguay 1018
15 Mexico 989
16 Colombia 986
17 2 Netherlands 981
18 3 Wales 953
19 1 Italy 951
20 2 Croatia 945
*Change from 17 May 2018
Complete rankings at FIFA.com
The FIFA World Ranking is a ranking system for men's national teams in association football, currently led by Germany.[1] The teams of the member nations of FIFA, football's world governing body, are ranked based on their game results with the most successful teams being ranked highest. The rankings were introduced in December 1992, and eight teams (Argentina, Belgium, Brazil, France, Germany, Italy, the Netherlands and Spain) have held the top position, of which Brazil have spent the longest ranked first.
A points system is used, with points being awarded based on the results of all FIFA-recognised full international matches.
The ranking system has been revamped on several occasions, generally responding to criticism that the preceding calculation method did not effectively reflect the relative strengths of the national teams. The latest calculation method will be used for the first time starting on 16 August 2018 and reflect matches held during the 2018 FIFA World Cup.
Alternative systems have been devised, such as the World Football Elo Ratings, based on the Elo rating system used in chess and Go, ranking teams.
Contents
1 History
1.1 1993–1998 calculation method
1.2 1999–2006 calculation method
1.3 2006–2018 calculation method
1.4 2018 ranking system update
2 Rank leaders
3 Uses of the rankings
4 Criticism
5 Current calculation method
6 Awards
6.1 Team of the Year
6.1.1 Performances by countries
6.2 Best Mover of the Year
7 Ranking schedule
8 See also
9 References
10 External links
History[edit]
In December 1992, FIFA first published a listing in rank order of its member associations to provide a basis for comparison of the relative strengths of these teams. From the following August, this list was more frequently updated, to be published most months.[2] Significant changes were implemented in January 1999 and again in July 2006, as a reaction to criticisms of the system.[3] Historical records of the rankings, such as listed at FIFA.com, reflect the method of calculation in use at the time, as the current method has not been applied retrospectively to rankings prior to July 2006. Membership of FIFA has expanded from 167 to 209 since the rankings began; all 209 members are currently included in the rankings.
1993–1998 calculation method[edit]
The ranking formula used from August 1993 until December 1998 was very simplistic and quickly became noticed for its lack of supporting factors. When the rankings were initially introduced, a team received one point for a draw or three for a victory in FIFA-recognised matches – much the same as a traditional league scoring system. This was a quite simplistic approach, however, and FIFA quickly realised that there were many factors affecting international matches.
1999–2006 calculation method[edit]
Main article: FIFA World Ranking system (1999–2006)
In January 1999, FIFA introduced a revised system of ranking calculation, incorporating many changes in response to criticism of inappropriate rankings. For the ranking all matches, their scores and importance were all recorded, and were used in the calculation procedure. Only matches for the senior men's national team were included. Separate ranking systems were used for other representative national sides such as women's and junior teams, for example the FIFA Women's World Rankings. The women's rankings were, and still are, based on a procedure which is a simplified version of the Football Elo Ratings.[4]
The major changes were as follows:
the point ranking was scaled up by a factor of ten
the method of calculation was changed to take into account factors including:
the number of goals scored or conceded
whether the match was played at home or away
the importance of a match or competition
regional strength
a fixed number of points were no longer necessarily awarded for a victory or a draw
match losers were able to earn points
Two new awards were introduced as part of the system:
Team of the Year
Best Mover of the Year
The changes made the ranking system more complex, but helped improve its accuracy by making it more comprehensive.
2006–2018 calculation method[edit]
Main article: FIFA World Ranking system (2006–2018)
FIFA announced that the ranking system would be updated following the 2006 World Cup. The evaluation period was cut from eight to four years, and a simpler method of calculation was used to determine rankings.[5] Goals scored and home or away advantage were no longer taken into account, and other aspects of the calculations, including the importance attributed to different types of match, were revised. The first set of revised rankings and the calculation methodology were announced on 12 July 2006.
This change was rooted at least in part in widespread criticism of the previous ranking system. Many football enthusiasts felt it was inaccurate, especially when compared to other ranking systems and that it was not sufficiently responsive to changes in the performance of individual teams.
2018 ranking system update[edit]
In September 2017, FIFA announced they were reviewing the ranking system and will decide after the end of the 2018 FIFA World Cup qualification if any changes are to be made to improve the ranking.[6]. FIFA announced on 10 June 2018 that the ranking system would be updated following the 2018 World Cup.
The calculation method to be adopted will be closely modeled after the Elo rating system and rankings of its member associations will be updated on a game-by-game basis. The weighting designated for each confederation for ranking purposes will be abolished.[7] However, the new methodology does not account for home or away games and margin of the victory, as Elo rankings.[8]
Rank leaders[edit]
FIFA World Ranking Leaders
When the system was introduced, Germany débuted as the top-ranked team following their extended period of dominance in which they had reached the three previous FIFA World Cup finals, winning one of them. Brazil took the lead in the run up to the 1994 FIFA World Cup after winning eight and losing only one of nine qualification matches, while on the way scoring twenty goals and conceding just four. Italy then led for a short time on the back of their own equally successful World Cup qualifying campaign, after which the top place was re-claimed by Germany.
Brazil's success in their lengthy qualifying campaign returned them to the lead for a brief period. Germany led again during the 1994 World Cup, until Brazil’s victory in that competition gave them a large lead that would stand up for nearly seven years, until they were surpassed by a strong France team that captured both the 1998 FIFA World Cup and the 2000 European Football Championship.
Success at the 2002 FIFA World Cup restored Brazil to the top position, where they remained until February 2007, when Italy returned to the top for the first time since 1993 following their 2006 FIFA World Cup win in Germany. Just one month later, Argentina replaced them, reaching the top for the first time, but Italy regained its place in April. After winning the Copa América 2007 in July, Brazil returned to the top, but were replaced by Italy in September and then Argentina in October.
In July 2008, Spain took over the lead for the first time, having won UEFA Euro 2008. Brazil began a sixth stint at the top of the rankings in July 2009 after winning the 2009 Confederations Cup, and Spain regained the title in November 2009 after winning every match in qualification for the 2010 FIFA World Cup.
In April 2010, Brazil returned to the top of the table. After winning the 2010 World Cup, Spain regained the top position and held it until August 2011, when the Netherlands reached the top spot for the first time,[9] only to relinquish it the following month.
In July 2014, Germany took over the lead once again, having won the 2014 FIFA World Cup. In July 2015, Argentina reached the top spot for the first time since 2008, after reaching both the 2014 FIFA World Cup Final, as well as the 2015 Copa America Final. In November 2015, Belgium became the leader in the FIFA rankings for the first time, after topping their Euro 2016 qualifying group. Belgium led the rankings until April 2016, when Argentina returned to the top. On April 6, 2017, Brazil returned to the No. 1 spot for the first time since just prior to the 2010 World Cup,[10] but Germany regained the top spot in July after winning the Confederations Cup.[11]
Uses of the rankings[edit]
The rankings are used by FIFA to rank the progression and current ability of the national football teams of its member nations, and claims that they create "a reliable measure for comparing national A-teams".[2] They are used as part of the calculation, or the entire grounds to seed competitions. In the 2010 FIFA World Cup qualification tournament, the rankings were used to seed the groups in the competitions involving CONCACAF members (using the May rankings), CAF (with the July set of data), and UEFA, using the specially postponed November 2007 ranking positions.
The October 2009 ranking was used to determine the seeds for the 2010 FIFA World Cup final draw.[12] The March 2011 ranking was used to seed the draw for the 2012 CAF Men's Pre-Olympic Tournament second qualifying round.[13]
The rankings are also used to determine the winners of the two annual awards national teams receive on the basis of their performance in the rankings.
The English Football Association uses the average of the last 24 months of rankings as one of the criteria for player work permits.[14]
Criticism[edit]
Since their introduction in 1992, the FIFA World Rankings have been the subject of much debate, particularly regarding the calculation procedure and the resulting disparity between generally perceived quality and world ranking of some teams. For example, Norway was ranked second in October 1993 and July–August 1995,[15] and the United States reached fourth in 2006, to the surprise of even their own players.[16] However, criticism regarding the unrealistic ranking continued even after the introduction of the new formula. Israel's climb to 15th in November 2008 surprised the Israeli press,[17][18][19] with the team missing a huge opportunity to break into the top 10 after conceding a last gasp equaliser against Latvia, a considerably worse ranked team.[20] Similarly, Belgium's rank of world number 1 in November 2015 was met with scepticism in the press, given that Belgium had only played in one tournament final stage in the past 13 years.[21]
Prior to July 2006, one of the main criticisms was that the rankings considered the performances of teams over an eight-year period, and that teams' ranking positions did not correlate to their recent performances.[15][22] This criticism has lessened somewhat with the introduction of a new formula, reflecting results over a four-year period, introduced in July 2006. However, problems remain, particularly that hosts of some major tournaments do not take part in qualifying rounds, and instead participate only in friendlies which offer fewer points.
This has been a factor which has resulted in 2014 World Cup hosts Brazil falling to a record low ranking of 22nd in the world.[23][24] The perceived flaws in the FIFA system have led to the creation of a number of alternative rankings from football statisticians,[15] including the World Football Elo Ratings and the rec.sport.soccer Statistics Foundation[citation needed] rankings.
The current system has been criticised for being open to abuse. National teams can maintain a higher average score by playing fewer friendly matches, particularly against weaker opponents.[25] For example, Switzerland were seeded for the 2014 World Cup draw, largely thanks to only playing three friendly matches in the previous year.[25] Switzerland would have reduced their rating if they had played any more friendlies, as their average score was greater than the maximum possible score from winning a friendly match against the top-ranked nation.[25]
Romania were one of the surprise seeds for the 2018 World Cup preliminary draw after taking advice from a consultant and playing only one friendly in the year before the draw.[26][27][28]
Lastly, the use of regional strength multiplier in the ranking determination formula has been accused of further reinforcing and perpetuating the bias for and against certain regions. FIFA's use of regional multiplier has been cited as a primary reason why most teams in AFC and CONCACAF have significantly lower FIFA rankings compared to their Elo rating based rankings.[29]
Current calculation method[edit]
On June 10, 2018, the new ranking system was approved by the FIFA Council. It is based on the Elo rating system and after each game points will be added to or subtracted from a team's rating according to the formula:
P = P before + I ( W − W e ) {\displaystyle P=P_{\text{before}}+I(W-W_{e})}
where:
Pbefore – the number of team's points before the game
I – the importance coefficient:
05 – friendlies played outside the International Match Calendar windows
10 – friendlies played within the International Match Calendar windows
15 – Nations League matches (group stage)
25 – Nations League matches (play-offs and finals)
25 – Confederations' final competitions qualifiers, FIFA World Cup qualifiers
35 – Confederations' final competitions matches (before quarter-finals)
40 – Confederations' final competitions matches (quarter-finals and later), FIFA Confederations Cup matches
50 – FIFA World Cup matches (before quarter-finals)
60 – FIFA World Cup matches (quarter-finals and later)
W – the result of the game: 0 – loss after regular or extra time, 0.5 – draw or loss in a penalty shootout, 0.75 - win in a penalty shootout, 1 – win after regular or extra time. If a game ends with a winner, but still requires a PSO (after a two-legged tie), it is considered as a regular game.
We – the expected result of the game:
W e = 1 10 − d r 600 + 1 {\displaystyle W_{e}={\frac {1}{10^{-{\frac {dr}{600}}}+1}}}
where dr is the difference between two teams' ratings before the game.
Negative points in knockout stages of final competitions will not affect teams' ratings.[30][31]
Awards[edit]
Each year FIFA hands out two awards to its member nations, based on their performance in the rankings. They are:
Team of the Year[edit]
Team of the Year is awarded to the team that finishes top of the FIFA World Ranking. Germany are the Team of the Year for the third time in the 24-year history of the rankings. Brazil hold the records for most consecutive wins (seven, between 1994 and 2000) and most wins overall (twelve). The table below shows the three best teams of each year.[32]
Year
First place
Second place
Third place
1993 Germany Italy Brazil
1994 Brazil Spain Sweden
1995 Brazil Germany Italy
1996 Brazil Germany France
1997 Brazil Germany Czech Republic
1998 Brazil France Germany
1999 Brazil Czech Republic France
2000 Brazil France Argentina
2001 France Argentina Brazil
2002 Brazil France Spain
2003 Brazil France Spain
2004 Brazil France Argentina
2005 Brazil Czech Republic Netherlands
2006 Brazil Italy Argentina
2007 Argentina Brazil Italy
2008 Spain Germany Netherlands
2009 Spain Brazil Netherlands
2010 Spain Netherlands Germany
2011 Spain Netherlands Germany
2012 Spain Germany Argentina
2013 Spain Germany Argentina
2014 Germany Argentina Colombia
2015 Belgium Argentina Spain
2016 Argentina Brazil Germany
2017 Germany Brazil Portugal
Performances by countries[edit]
Team
First place
Second place
Third place
Brazil 12 (1994, 1995, 1996, 1997, 1998, 1999, 2000, 2002, 2003, 2004, 2005, 2006) 4 (2007, 2009, 2016, 2017) 2 (1993, 2001)
Spain 6 (2008, 2009, 2010, 2011, 2012, 2013) 1 (1994) 3 (2002, 2003, 2015)
Germany 3 (1993, 2014, 2017) 6 (1995, 1996, 1997, 2008, 2012, 2013) 4 (1998, 2010, 2011, 2016)
Argentina 2 (2007, 2016) 3 (2001, 2014, 2015) 5 (2000, 2004, 2006, 2012, 2013)
France 1 (2001) 5 (1998, 2000, 2002, 2003, 2004) 2 (1996, 1999)
Belgium 1 (2015) 0 0
Netherlands 0 2 (2010, 2011) 3 (2005, 2008, 2009)
Italy 0 2 (1993, 2006) 2 (1995, 2007)
Czech Republic 0 2 (1999, 2005) 1 (1997)
Sweden 0 0 1 (1994)
Colombia 0 0 1 (2014)
Portugal 0 0 1 (2017)
Best Mover of the Year[edit]
The Best Mover of the Year was awarded to the team who made the best progress up the rankings over the course of the year. In the FIFA rankings, this is not simply the team that has risen the most places, but a calculation is performed in order to account for the fact that it becomes progressively harder to earn more points the higher up the rankings a team is.[2]
The calculation used is the number of points the team has at the end of the year (z) multiplied by the number of points it earned during the year (y). The team with the highest index on this calculation received the award. The table below shows the top three best movers from each year.[33][34]
The award has not been an official part of the awards since 2006.
Year
First place
Second place
Third place
1993 Colombia Portugal Morocco
1994 Croatia Brazil Uzbekistan
1995 Jamaica Trinidad and Tobago Czech Republic
1996 South Africa Paraguay Canada
1997 Yugoslavia Bosnia and Herzegovina Iran
1998 Croatia France Argentina
1999 Slovenia Cuba Uzbekistan
2000 Nigeria Honduras Cameroon
2001 Costa Rica Australia Honduras
2002 Senegal Wales Brazil
2003 Bahrain Oman Turkmenistan
2004 China PR Uzbekistan Ivory Coast
2005 Ghana Ethiopia Switzerland
2006 Italy Germany France
While an official award has not been made for movements since 2006, FIFA has released a list of the 'Best Movers' in the rankings since 2007.[35]
An example of the informal on-going "Mover of the Year" award is the recognition made by FIFA to Colombia in 2012 in an official press release.[36] However, the calculation methodology had changed to the difference in ranking points over the course of the year (rather than the methodology used in the official award from 1993 to 2006). The results for latter years are based on a similar methodology.
Year
Best mover
Second best
Third best
2007 Mozambique Norway New Caledonia
2008 Spain Montenegro Russia
2009 Brazil Algeria Slovenia
2010 Netherlands Montenegro Botswana
2011 Wales Sierra Leone Bosnia and Herzegovina
2012 Colombia Ecuador Mali
2013[37] Ukraine Armenia United States
2014[38] Germany Slovakia Belgium
2015[39] Turkey Hungary Nicaragua
2016[40] France Peru Poland
2017 Denmark Sweden Bolivia
Ranking schedule[edit]
Rankings are published monthly, usually on a Thursday. The deadline for the matches to be considered is usually the Thursday prior to the release date, but after major tournaments, all games up to the final are included.[41] The 19 July 2018 release was cancelled following the new calculation method implementation.
Rankings schedule 2018
Release date
18 January
15 February
15 March
12 April
17 May
7 June
16 August
20 September
25 October
29 November
20 December
See also[edit]
FIFA Women's World Rankings
World Football Elo Ratings
Seeding for 2006 FIFA World Cup
Statistical association football predictions
References[edit]
^ Jump up to: a b "The FIFA/Coca-Cola World Ranking – Ranking Table". FIFA. Retrieved 7 June 2018.
^ Jump up to: a b c "FIFA/Coca-Cola World Ranking Procedure". FIFA. Retrieved 28 March 2008.
Jump up ^ "Revised FIFA/Coca-Cola World Ranking". FIFA. 6 July 2006. Archived from the original on 12 June 2008. Retrieved 28 March 2008.
Jump up ^ "Women's World Ranking Procedure". FIFA. Retrieved 28 March 2008.
Jump up ^ "Great expectations". FIFA. 17 May 2006. Archived from the original on 12 June 2008. Retrieved 28 March 2008.
Jump up ^ Ben Rumsby (4 September 2017). "Fifa to review rankings system after friendlies harm England's World Cup seeding". The Telegraph. Retrieved 4 October 2017.
Jump up ^ Price, Steve (11 June 2018). "How FIFA's New Ranking System Will Change International Soccer". Forbes. Retrieved 12 June 2018.
Jump up ^ "FIFA World Ranking gets overhaul to stop manipulation for draw seedings". 11 June 2018.
Jump up ^ "FIFA Ranking: Netherlands dethrone Spain". Football-Rankings.info. 12 August 2011. Retrieved 24 August 2011.
Jump up ^ "Brazil back on top of FIFA Ranking, Netherlands slump to record low". ESPN. 29 March 2017. Retrieved 29 March 2017.
Jump up ^ "Fifa rankings: Northern Ireland reach highest point as Germany lead the way". BBC news. 6 July 2017. Retrieved 8 July 2017.
Jump up ^ "FIFA Organising Committee approves Final Draw procedure". FIFA. 2 December 2009. Archived from the original on 25 December 2011. Retrieved 13 April 2011.
Jump up ^ "Olympics – updated draw procedure, with teams". CAFOnline.com. Confederation of African Football. 12 April 2011. Retrieved 13 April 2011.
Jump up ^ "POINTS BASED SYSTEM GOVERNING BODY ENDORSEMENT REQUIREMENTS FOR PLAYERS 2017/2018 SEASON" (pdf). English Football Association. Retrieved 29 November 2017.
^ Jump up to: a b c "The world rankings riddle". BBC Sport. 21 December 2000.
Jump up ^ "FIFA adapting new world rankings" (PDF). Associated Press. 2 June 2006. Retrieved 23 July 2017.
Jump up ^ "FIFA chief praises IFA's work". The Jerusalem Post. 29 August 2008. Archived from the original on 27 April 2011. Retrieved 13 February 2009. Blatter was also asked about Israel's unrealistic FIFA ranking. Despite failing to qualify for a major tournament in 38 years, the national team is currently ranked number 16 in the world, ahead of such teams as Greece, Sweden and Denmark, and just two places below England.
Jump up ^ "Soccer: Quirky FIFA rankings system sees Israel climb to #15". The Jerusalem Post. 13 November 2008. Archived from the original on 16 November 2008. Retrieved 13 February 2009. Israel continued its baffling rise up the FIFA rankings on Wednesday, climbing one more position to a best-ever 15th in the world. Amazingly, the national team, which hasn't reached a major tournament since the 1970 World Cup, is ahead of such teams as Greece (18), Nigeria (22), Sweden (29), Scotland (33), Denmark (34) and the Republic of Ireland (36)
Jump up ^ "The Last Word: It's time to get rid of the FIFA rankings". The Jerusalem Post. 14 November 2008. Archived from the original on 16 November 2008. Retrieved 13 February 2009. So the news this week that Israel has risen another place to number 15 in the latest FIFA rankings, above Mexico (25), Nigeria (22), the US (24) and Colombia (40), only serves to underline the futility of the ranking system.
Jump up ^ "FIFA Ranking: November 2008 preview (II)". Football-Rankings.info. 20 October 2008. Retrieved 13 February 2009. Israel will improve (again!) their best ever ranking, climbing one place to 15th. With a win (instead of a draw) against Latvia they would have been tied for 8th place with Russia.
Jump up ^ Alistair Tweedale (13 October 2015). "Belgium rise to No1 in Fifa world rankings after they beat Israel – despite playing one tournament in 13 years". The Telegraph. Retrieved 14 November 2015.
Jump up ^ "FIFA Rankings". Travour.com. 1 June 2006. Archived from the original on 14 June 2006.
Jump up ^ Cummings, Michael. "FIFA World Rankings Place Brazil 18th, Reinforce Flawed Nature of System". Retrieved 21 May 2013.
Jump up ^ Freitas, Paulo. "Scolari blames FIFA for Brazil's poor ranking". Retrieved 16 June 2013.
^ Jump up to: a b c Burnton, Simon (17 October 2013). "With a bit of planning England could have been seeded ahead of the Swiss". The Guardian. Retrieved 15 June 2015.
Jump up ^ "Romania, Wales set to be surprise World Cup top seeds". USA Today. Associated Press. 23 July 2015. Retrieved 29 July 2015. Romania took advice from a consultant who decoded some mysteries of a ranking system used by FIFA since 1992.
Jump up ^ Gavril, Victor (5 July 2015). "Burleanu: "Am consultat specialiști în clasamentul FIFA." Cum a urcat România în TOP 10 mondial" [Burleanu: "We have consulted FIFA ranking specialists." How Romania climbed in the World Top 10]. România Liberă (in Romanian). Retrieved 29 July 2015.
Jump up ^ "2018 FIFA World Cup: Preliminary draw pots for UEFA". Football-Rankings.info. 26 June 2015. Retrieved 29 July 2015. I've told Romanian FA officials in the spring of 2014 to avoid playing friendlies. Surprisingly, they heeded my advice.
Jump up ^ "Soccer rankings: farewell FIFA, hello Elo?". Retrieved 2016-07-07.
Jump up ^ Revision of the FIFA / Coca-Cola World Ranking
Jump up ^ Revision of the FIFA / Coca - Cola World Ranking
Jump up ^ Team of the Year Award 2010 Archived 18 December 2010 at the Wayback Machine. on the FIFA website
Jump up ^ Best Mover of the Year Archived 20 July 2008 at the Wayback Machine. on the FIFA website
Jump up ^ Spain on top and Wales highest climber on the FIFA website
Jump up ^ Top Team and the Best Mover of the Year on the FIFA website (PDF)
Jump up ^ Spain finish 2012 on top, Colombia in fifth. FIFA.com. Retrieved on 12 August 2013.
Jump up ^ "Spain on top, Ukraine highest climber". fifa.com. 19 December 2013. Retrieved 19 December 2013.
Jump up ^ "Germany conquer 2014, Belgium, Slovakia impress". fifa.com. 18 December 2014. Retrieved 18 December 2014.
Jump up ^ "Belgium and Turkey claim awards, Hungary return". fifa.com. 3 December 2015. Archived from the original on 21 July 2016. Retrieved 15 December 2015.
Jump up ^ "Argentina and France take Ranking awards". fifa.com. 22 December 2016. Archived from the original on 22 December 2016. Retrieved 12 December 2016.
Jump up ^ Men's Ranking Procedure on the FIFA website
External links[edit]
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text:
Naive Bayes classifier
From Wikipedia, the free encyclopedia
Jump to: navigation, search
In machine learning, naive Bayes classifiers are a family of simple "probabilistic classifiers" based on applying Bayes' theorem with strong (naive) independence assumptions between the features.
Naive Bayes has been studied extensively since the 1950s. It was introduced under a different name into the text retrieval community in the early 1960s,[1]:488 and remains a popular (baseline) method for text categorization, the problem of judging documents as belonging to one category or the other (such as spam or legitimate, sports or politics, etc.) with word frequencies as the features. With appropriate pre-processing, it is competitive in this domain with more advanced methods including support vector machines.[2] It also finds application in automatic medical diagnosis.[3]
Naive Bayes classifiers are highly scalable, requiring a number of parameters linear in the number of variables (features/predictors) in a learning problem. Maximum-likelihood training can be done by evaluating a closed-form expression,[1]:718 which takes linear time, rather than by expensive iterative approximation as used for many other types of classifiers.
In the statistics and computer science literature, naive Bayes models are known under a variety of names, including simple Bayes and independence Bayes.[4] All these names reference the use of Bayes' theorem in the classifier's decision rule, but naive Bayes is not (necessarily) a Bayesian method.[1][4]
Contents
[hide]
1 Introduction
2 Probabilistic model
2.1 Constructing a classifier from the probability model
3 Parameter estimation and event models
3.1 Gaussian naive Bayes
3.2 Multinomial naive Bayes
3.3 Bernoulli naive Bayes
3.4 Semi-supervised parameter estimation
4 Discussion
4.1 Relation to logistic regression
5 Examples
5.1 Sex classification
5.1.1 Training
5.1.2 Testing
5.2 Document classification
6 See also
7 References
7.1 Further reading
8 External links
Introduction[edit]
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. It 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.
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.
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.[5] 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.[6]
An advantage of naive Bayes is that it only requires a small number of training data to estimate the parameters necessary for classification.[citation needed]
Probabilistic model[edit]
Abstractly, naive Bayes is a conditional probability model: given a problem instance to be classified, represented by a vector x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})} representing some n features (independent variables), it assigns to this instance probabilities
p ( C k ∣ x 1 , … , x n ) {\displaystyle p(C_{k}\mid x_{1},\dots ,x_{n})\,}
for each of K possible outcomes or classes C k {\displaystyle C_{k}} .[7]
The problem with the above formulation is that if the number of features n is large or if 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, the conditional probability can be decomposed as
p ( C k ∣ x ) = p ( C k ) p ( x ∣ C k ) p ( x ) {\displaystyle p(C_{k}\mid \mathbf {x} )={\frac {p(C_{k})\ p(\mathbf {x} \mid C_{k})}{p(\mathbf {x} )}}\,}
In plain English, using Bayesian probability terminology, the above equation can be written as
posterior = prior × likelihood evidence {\displaystyle {\mbox{posterior}}={\frac {{\mbox{prior}}\times {\mbox{likelihood}}}{\mbox{evidence}}}\,}
In practice, there is interest only in the numerator of that fraction, because the denominator does not depend on C {\displaystyle C} and the values of the features x i {\displaystyle x_{i}} are given, so that the denominator is effectively constant. The numerator is equivalent to the joint probability model
p ( C k , x 1 , … , x n ) {\displaystyle p(C_{k},x_{1},\dots ,x_{n})\,}
which can be rewritten as follows, using the chain rule for repeated applications of the definition of conditional probability:
p ( C k , x 1 , … , x n ) = p ( x 1 , … , x n , C k ) = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 , … , x n , C k ) = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 ∣ x 3 , … , x n , C k ) p ( x 3 , … , x n , C k ) = … = p ( x 1 ∣ x 2 , … , x n , C k ) p ( x 2 ∣ x 3 , … , x n , C k ) … p ( x n − 1 ∣ x n , C k ) p ( x n ∣ C k ) p ( C k ) {\displaystyle {\begin{aligned}p(C_{k},x_{1},\dots ,x_{n})&=p(x_{1},\dots ,x_{n},C_{k})\\&=p(x_{1}\mid x_{2},\dots ,x_{n},C_{k})p(x_{2},\dots ,x_{n},C_{k})\\&=p(x_{1}\mid x_{2},\dots ,x_{n},C_{k})p(x_{2}\mid x_{3},\dots ,x_{n},C_{k})p(x_{3},\dots ,x_{n},C_{k})\\&=\dots \\&=p(x_{1}\mid x_{2},\dots ,x_{n},C_{k})p(x_{2}\mid x_{3},\dots ,x_{n},C_{k})\dots p(x_{n-1}\mid x_{n},C_{k})p(x_{n}\mid C_{k})p(C_{k})\\\end{aligned}}}
Now the "naive" conditional independence assumptions come into play: assume that each feature x i {\displaystyle x_{i}} is conditionally independent of every other feature x j {\displaystyle x_{j}} for j ≠ i {\displaystyle j\neq i} , given the category C k {\displaystyle C_{k}} . This means that
p ( x i ∣ x i + 1 , … , x n , C k ) = p ( x i ∣ C k ) {\displaystyle p(x_{i}\mid x_{i+1},\dots ,x_{n},C_{k})=p(x_{i}\mid C_{k})\,} .
Thus, the joint model can be expressed as
p ( C k ∣ x 1 , … , x n ) ∝ p ( C k , x 1 , … , x n ) = = p ( C k ) p ( x 1 ∣ C k ) p ( x 2 ∣ C k ) p ( x 3 ∣ C k ) ⋯ = p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) . {\displaystyle {\begin{aligned}p(C_{k}\mid x_{1},\dots ,x_{n})&\varpropto p(C_{k},x_{1},\dots ,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{aligned}}}
Where ∝ {\displaystyle \varpropto } denotes proportionality.
This means that under the above independence assumptions, the conditional distribution over the class variable C {\displaystyle C} is:
p ( C k ∣ x 1 , … , x n ) = 1 Z p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) {\displaystyle p(C_{k}\mid x_{1},\dots ,x_{n})={\frac {1}{Z}}p(C_{k})\prod _{i=1}^{n}p(x_{i}\mid C_{k})}
where the evidence Z = p ( x ) = ∑ k p ( C k ) p ( x ∣ C k ) {\displaystyle Z=p(\mathbf {x} )=\sum _{k}p(C_{k})\ p(\mathbf {x} \mid C_{k})} is a scaling factor dependent only on x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} , that is, a constant if the values of the feature variables are known.
Constructing a classifier from the probability model[edit]
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, a Bayes classifier, is the function that assigns a class label y ^ = C k {\displaystyle {\hat {y}}=C_{k}} for some k as follows:
y ^ = argmax k ∈ { 1 , … , K } p ( C k ) ∏ i = 1 n p ( x i ∣ C k ) . {\displaystyle {\hat {y}}={\underset {k\in \{1,\dots ,K\}}{\operatorname {argmax} }}\ p(C_{k})\displaystyle \prod _{i=1}^{n}p(x_{i}\mid C_{k}).}
Parameter estimation and event models[edit]
A class's 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.[8]
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.[9][10]
Gaussian naive Bayes[edit]
When dealing with continuous data, a typical assumption is that the continuous values associated with each class are distributed according to a Gaussian distribution. For example, suppose the training data contains a continuous attribute, x {\displaystyle x} . We first segment the data by the class, and then compute the mean and variance of x {\displaystyle x} in each class. Let μ k {\displaystyle \mu _{k}} be the mean of the values in x {\displaystyle x} associated with class Ck, and let σ k 2 {\displaystyle \sigma _{k}^{2}} be the variance of the values in x {\displaystyle x} associated with class Ck. Suppose we have collected some observation value v {\displaystyle v} . Then, the probability distribution of v {\displaystyle v} given a class C k {\displaystyle C_{k}} , p ( x = v ∣ C k ) {\displaystyle p(x=v\mid C_{k})} , can be computed by plugging v {\displaystyle v} into the equation for a Normal distribution parameterized by μ k {\displaystyle \mu _{k}} and σ k 2 {\displaystyle \sigma _{k}^{2}} . That is,
p ( x = v ∣ C k ) = 1 2 π σ k 2 e − ( v − μ k ) 2 2 σ k 2 {\displaystyle p(x=v\mid C_{k})={\frac {1}{\sqrt {2\pi \sigma _{k}^{2}}}}\,e^{-{\frac {(v-\mu _{k})^{2}}{2\sigma _{k}^{2}}}}}
Another common technique for handling continuous values is to use binning to discretize the feature values, to obtain a new set of Bernoulli-distributed features; some literature in fact suggests that this is necessary to apply naive Bayes, but it is not, and the discretization may throw away discriminative information.[4]
Multinomial naive Bayes[edit]
With a multinomial event model, samples (feature vectors) represent the frequencies with which certain events have been generated by a multinomial ( p 1 , … , p n ) {\displaystyle (p_{1},\dots ,p_{n})} where p i {\displaystyle p_{i}} is the probability that event i occurs (or K such multinomials in the multiclass case). A feature vector x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\dots ,x_{n})} is then a histogram, with x i {\displaystyle x_{i}} counting the number of times event i 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 x is given by
p ( x ∣ C k ) = ( ∑ i x i ) ! ∏ i x i ! ∏ i p k i x i {\displaystyle p(\mathbf {x} \mid C_{k})={\frac {(\sum _{i}x_{i})!}{\prod _{i}x_{i}!}}\prod _{i}{p_{ki}}^{x_{i}}}
The multinomial naive Bayes classifier becomes a linear classifier when expressed in log-space:[2]
log  p ( C k ∣ x ) ∝ log  ( p ( C k ) ∏ i = 1 n p k i x i ) = log  p ( C k ) + ∑ i = 1 n x i ⋅ log  p k i = b + w k ⊤ x {\displaystyle {\begin{aligned}\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{aligned}}}
where b = log  p ( C k ) {\displaystyle b=\log p(C_{k})} and w k i = log  p k i {\displaystyle w_{ki}=\log p_{ki}} .
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. 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.[2]
Bernoulli naive Bayes[edit]
In the multivariate Bernoulli event model, features are independent booleans (binary variables) describing inputs. Like the multinomial model, this model is popular for document classification tasks,[9] where binary term occurrence features are used rather than term frequencies. If x i {\displaystyle x_{i}} is a boolean expressing the occurrence or absence of the i'th term from the vocabulary, then the likelihood of a document given a class C k {\displaystyle C_{k}} is given by[9]
p ( x ∣ C k ) = ∏ i = 1 n p k i x i ( 1 − p k i ) ( 1 − x i ) {\displaystyle p(\mathbf {x} \mid C_{k})=\prod _{i=1}^{n}p_{ki}^{x_{i}}(1-p_{ki})^{(1-x_{i})}}
where p k i {\displaystyle p_{ki}} is the probability of class C k {\displaystyle C_{k}} generating the term x i {\displaystyle x_{i}} . This event model is especially popular for classifying short texts. It has the benefit of explicitly modelling the absence of terms. Note that a naive Bayes classifier with a Bernoulli event model is not the same as a multinomial NB classifier with frequency counts truncated to one.
Semi-supervised parameter estimation[edit]
Given a way to train a naive Bayes classifier from labeled data, it's possible to construct a semi-supervised training algorithm that can learn from a combination of labeled and unlabeled data by running the supervised learning algorithm in a loop:[11]
Given a collection D = L ⊎ U {\displaystyle D=L\uplus U} of labeled samples L and unlabeled samples U, start by training a naive Bayes classifier on L.
Until convergence, do:
Predict class probabilities P ( C ∣ x ) {\displaystyle P(C\mid x)} for all examples x in D {\displaystyle D} .
Re-train the model based on the probabilities (not the labels) predicted in the previous step.
Convergence is determined based on improvement to the model likelihood P ( D ∣ θ ) {\displaystyle P(D\mid \theta )} , where θ {\displaystyle \theta } denotes the parameters of the naive Bayes model.
This training algorithm is an instance of the more general expectation–maximization algorithm (EM): the prediction step inside the loop is the E-step of EM, while the re-training of naive Bayes is the M-step. The algorithm is formally justified by the assumption that the data are generated by a mixture model, and the components of this mixture model are exactly the classes of the classification problem.[11]
Discussion[edit]
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,[12] 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.[3] Other reasons for the observed success of the naive Bayes classifier are discussed in the literature cited below.
Relation to logistic regression[edit]
This section needs expansion. You can help by adding to it. (August 2014)
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 p ( C , x ) {\displaystyle p(C,\mathbf {x} )} , while logistic regression fits the same probability model to optimize the conditional p ( C ∣ x ) {\displaystyle p(C\mid \mathbf {x} )} .[13]
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 C 1 {\displaystyle C_{1}} if the odds of p ( C 1 ∣ x ) {\displaystyle p(C_{1}\mid \mathbf {x} )} exceed those of p ( C 2 ∣ x ) {\displaystyle p(C_{2}\mid \mathbf {x} )} ". Expressing this in log-space gives:
log  p ( C 1 ∣ x ) p ( C 2 ∣ x ) = log  p ( C 1 ∣ x ) − log  p ( C 2 ∣ x ) > 0 {\displaystyle \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}
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 b + w ⊤ x > 0 {\displaystyle b+\mathbf {w} ^{\top }x>0} . Obtaining the probabilities is then a matter of applying the logistic function to b + w ⊤ x {\displaystyle b+\mathbf {w} ^{\top }x} , 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.[13]
Examples[edit]
Sex classification[edit]
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[edit]
Example training set below.
Person
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 unbiased sample variances):
Person
mean (height)
variance (height)
mean (weight)
variance (weight)
mean (foot size)
variance (foot size)
male 5.855 3.5033*10-02 176.25 1.2292*10+02 11.25 9.1667*10-01
female 5.4175 9.7225*10-02 132.5 5.5833*10+02 7.5 1.6667
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[edit]
Below is a sample to be classified as male or female.
Person
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
posterior (male) = P ( male ) p ( height ∣ male ) p ( weight ∣ male ) p ( foot size ∣ male ) e v i d e n c e {\displaystyle {\text{posterior (male)}}={\frac {P({\text{male}})\,p({\text{height}}\mid {\text{male}})\,p({\text{weight}}\mid {\text{male}})\,p({\text{foot size}}\mid {\text{male}})}{evidence}}}
For the classification as female the posterior is given by
posterior (female) = P ( female ) p ( height ∣ female ) p ( weight ∣ female ) p ( foot size ∣ female ) e v i d e n c e {\displaystyle {\text{posterior (female)}}={\frac {P({\text{female}})\,p({\text{height}}\mid {\text{female}})\,p({\text{weight}}\mid {\text{female}})\,p({\text{foot size}}\mid {\text{female}})}{evidence}}}
The evidence (also termed normalizing constant) may be calculated:
evidence = P ( male ) p ( height ∣ male ) p ( weight ∣ male ) p ( foot size ∣ male ) + P ( female ) p ( height ∣ female ) p ( weight ∣ female ) p ( foot size ∣ female ) {\displaystyle {\begin{aligned}{\text{evidence}}=P({\text{male}})\,p({\text{height}}\mid {\text{male}})\,p({\text{weight}}\mid {\text{male}})\,p({\text{foot size}}\mid {\text{male}})\\+P({\text{female}})\,p({\text{height}}\mid {\text{female}})\,p({\text{weight}}\mid {\text{female}})\,p({\text{foot size}}\mid {\text{female}})\end{aligned}}}
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.
P ( male ) = 0.5 {\displaystyle P({\text{male}})=0.5}
p ( height ∣ male ) = 1 2 π σ 2 exp  ( − ( 6 − μ ) 2 2 σ 2 ) ≈ 1.5789 {\displaystyle p({\text{height}}\mid {\text{male}})={\frac {1}{\sqrt {2\pi \sigma ^{2}}}}\exp \left({\frac {-(6-\mu )^{2}}{2\sigma ^{2}}}\right)\approx 1.5789} ,
where μ = 5.855 {\displaystyle \mu =5.855} and σ 2 = 3.5033 ⋅ 10 − 2 {\displaystyle \sigma ^{2}=3.5033\cdot 10^{-2}} 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.
p ( weight ∣ male ) = 5.9881 ⋅ 10 − 6 {\displaystyle p({\text{weight}}\mid {\text{male}})=5.9881\cdot 10^{-6}}
p ( foot size ∣ male ) = 1.3112 ⋅ 10 − 3 {\displaystyle p({\text{foot size}}\mid {\text{male}})=1.3112\cdot 10^{-3}}
posterior numerator (male) = their product = 6.1984 ⋅ 10 − 9 {\displaystyle {\text{posterior numerator (male)}}={\text{their product}}=6.1984\cdot 10^{-9}}
P ( female ) = 0.5 {\displaystyle P({\text{female}})=0.5}
p ( height ∣ female ) = 2.2346 ⋅ 10 − 1 {\displaystyle p({\text{height}}\mid {\text{female}})=2.2346\cdot 10^{-1}}
p ( weight ∣ female ) = 1.6789 ⋅ 10 − 2 {\displaystyle p({\text{weight}}\mid {\text{female}})=1.6789\cdot 10^{-2}}
p ( foot size ∣ female ) = 2.8669 ⋅ 10 − 1 {\displaystyle p({\text{foot size}}\mid {\text{female}})=2.8669\cdot 10^{-1}}
posterior numerator (female) = their product = 5.3778 ⋅ 10 − 4 {\displaystyle {\text{posterior numerator (female)}}={\text{their product}}=5.3778\cdot 10^{-4}}
Since posterior numerator is greater in the female case, we predict the sample is female.
Document classification[edit]
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 modeled 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
p ( w i ∣ C ) {\displaystyle p(w_{i}\mid C)\,}
(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 w i {\displaystyle w_{i}} , given a class C, is
p ( D ∣ C ) = ∏ i p ( w i ∣ C ) {\displaystyle p(D\mid C)=\prod _{i}p(w_{i}\mid C)\,}
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 p ( C ∣ D ) {\displaystyle p(C\mid D)\,} ?
Now by definition
p ( D ∣ C ) = p ( D ∩ C ) p ( C ) {\displaystyle p(D\mid C)={p(D\cap C) \over p(C)}}
and
p ( C ∣ D ) = p ( D ∩ C ) p ( D ) {\displaystyle p(C\mid D)={p(D\cap C) \over p(D)}}
Bayes' theorem manipulates these into a statement of probability in terms of likelihood.
p ( C ∣ D ) = p ( C ) p ( D ∣ C ) p ( D ) {\displaystyle p(C\mid D)={\frac {p(C)\,p(D\mid C)}{p(D)}}}
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;
p ( D ∣ S ) = ∏ i p ( w i ∣ S ) {\displaystyle p(D\mid S)=\prod _{i}p(w_{i}\mid S)\,}
and
p ( D ∣ ¬ S ) = ∏ i p ( w i ∣ ¬ S ) {\displaystyle p(D\mid \neg S)=\prod _{i}p(w_{i}\mid \neg S)\,}
Using the Bayesian result above, we can write:
p ( S ∣ D ) = p ( S ) p ( D ) ∏ i p ( w i ∣ S ) {\displaystyle p(S\mid D)={p(S) \over p(D)}\,\prod _{i}p(w_{i}\mid S)}
p ( ¬ S ∣ D ) = p ( ¬ S ) p ( D ) ∏ i p ( w i ∣ ¬ S ) {\displaystyle p(\neg S\mid D)={p(\neg S) \over p(D)}\,\prod _{i}p(w_{i}\mid \neg S)}
Dividing one by the other gives:
p ( S ∣ D ) p ( ¬ S ∣ D ) = p ( S ) ∏ i p ( w i ∣ S ) p ( ¬ S ) ∏ i p ( w i ∣ ¬ S ) {\displaystyle {p(S\mid D) \over p(\neg S\mid D)}={p(S)\,\prod _{i}p(w_{i}\mid S) \over p(\neg S)\,\prod _{i}p(w_{i}\mid \neg S)}}
Which can be re-factored as:
p ( S ∣ D ) p ( ¬ S ∣ D ) = p ( S ) p ( ¬ S ) ∏ i p ( w i ∣ S ) p ( w i ∣ ¬ S ) {\displaystyle {p(S\mid D) \over p(\neg S\mid D)}={p(S) \over p(\neg S)}\,\prod _{i}{p(w_{i}\mid S) \over p(w_{i}\mid \neg S)}}
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:
ln  p ( S ∣ D ) p ( ¬ S ∣ D ) = ln  p ( S ) p ( ¬ S ) + ∑ i ln  p ( w i ∣ S ) p ( w i ∣ ¬ S ) {\displaystyle \ln {p(S\mid D) \over p(\neg S\mid D)}=\ln {p(S) \over p(\neg S)}+\sum _{i}\ln {p(w_{i}\mid S) \over p(w_{i}\mid \neg S)}}
(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.)
Finally, the document can be classified as follows. It is spam if p ( S ∣ D ) > p ( ¬ S ∣ D ) {\displaystyle p(S\mid D)>p(\neg S\mid D)} (i. e., ln  p ( S ∣ D ) p ( ¬ S ∣ D ) > 0 {\displaystyle \ln {p(S\mid D) \over p(\neg S\mid D)}>0} ), otherwise it is not spam.
See also[edit]
AODE
Bayesian spam filtering
Bayesian network
Random naive Bayes
Linear classifier
Logistic regression
Perceptron
Take-the-best heuristic
References[edit]
This article includes a list of references, but its sources remain unclear because it has insufficient inline citations. Please help to improve this article by introducing more precise citations. (May 2009) (Learn how and when to remove this template message)
^ Jump up to: a b c Russell, Stuart; Norvig, Peter (2003) [1995]. Artificial Intelligence: A Modern Approach (2nd ed.). Prentice Hall. ISBN 978-0137903955.
^ Jump up to: a b c Rennie, J.; Shih, L.; Teevan, J.; Karger, D. (2003). Tackling the poor assumptions of Naive Bayes classifiers (PDF). ICML.
^ Jump up to: a b Rish, Irina (2001). An empirical study of the naive Bayes classifier (PDF). IJCAI Workshop on Empirical Methods in AI.
^ Jump up to: a b c Hand, D. J.; Yu, K. (2001). "Idiot's Bayes — not so stupid after all?". International Statistical Review. 69 (3): 385–399. doi:10.2307/1403452. ISSN 0306-7734.
Jump up ^ Zhang, Harry. The Optimality of Naive Bayes (PDF). FLAIRS2004 conference.
Jump up ^ Caruana, R.; Niculescu-Mizil, A. (2006). An empirical comparison of supervised learning algorithms. Proc. 23rd International Conference on Machine Learning. CiteSeerX 10.1.1.122.5901  .
Jump up ^ Narasimha Murty, M.; Susheela Devi, V. (2011). Pattern Recognition: An Algorithmic Approach. ISBN 0857294946.
Jump up ^ John, George H.; Langley, Pat (1995). Estimating Continuous Distributions in Bayesian Classifiers. Proc. Eleventh Conf. on Uncertainty in Artificial Intelligence. Morgan Kaufmann. pp. 338–345.
^ Jump up to: a b c McCallum, Andrew; Nigam, Kamal (1998). A comparison of event models for Naive Bayes text classification (PDF). AAAI-98 workshop on learning for text categorization. 752.
Jump up ^ Metsis, Vangelis; Androutsopoulos, Ion; Paliouras, Georgios (2006). Spam filtering with Naive Bayes—which Naive Bayes?. Third conference on email and anti-spam (CEAS). 17.
^ Jump up to: a b Nigam, Kamal; McCallum, Andrew; Thrun, Sebastian; Mitchell, Tom (2000). "Learning to classify text from labeled and unlabeled documents using EM" (PDF). Machine Learning.
Jump up ^ Niculescu-Mizil, Alexandru; Caruana, Rich (2005). Predicting good probabilities with supervised learning (PDF). ICML. doi:10.1145/1102351.1102430.
^ Jump up to: a b Ng, Andrew Y.; Jordan, Michael I. (2002). On discriminative vs. generative classifiers: A comparison of logistic regression and naive Bayes. NIPS. 14.
Further reading[edit]
Domingos, Pedro; Pazzani, Michael (1997). "On the optimality of the simple Bayesian classifier under zero-one loss". Machine Learning. 29: 103–137.
Webb, G. I.; Boughton, J.; Wang, Z. (2005). "Not So Naive Bayes: Aggregating One-Dependence Estimators". Machine Learning. Springer. 58 (1): 5–24. doi:10.1007/s10994-005-4258-6.
Mozina, M.; Demsar, J.; Kattan, M.; Zupan, B. (2004). Nomograms for Visualization of Naive Bayesian Classifier (PDF). Proc. PKDD-2004. pp. 337–348.
Maron, M. E. (1961). "Automatic Indexing: An Experimental Inquiry". Journal of the ACM. 8 (3): 404–417. doi:10.1145/321075.321084.
Minsky, M. (1961). Steps toward Artificial Intelligence. Proc. IRE. 49. pp. 8–30.
External links[edit]
Book Chapter: Naive Bayes text classification, Introduction to Information Retrieval
Naive Bayes for Text Classification with Unbalanced Classes
Benchmark results of Naive Bayes implementations
Hierarchical Naive Bayes Classifiers for uncertain data (an extension of the Naive Bayes classifier).
Software
Naive Bayes classifiers are available in many general-purpose machine learning and NLP packages, including Apache Mahout, Mallet, NLTK, Orange, scikit-learn and 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.
An interactive Microsoft Excel spreadsheet Naive Bayes implementation using VBA (requires enabled macros) with viewable source code.
jBNC - Bayesian Network Classifier Toolbox
Statistical Pattern Recognition Toolbox for Matlab.
ifile - the first freely available (Naive) Bayesian mail/spam filter
NClassifier - NClassifier is a .NET library that supports text classification and text summarization. It is a port of Classifier4J.
Classifier4J - Classifier4J is a Java library designed to do text classification. It comes with an implementation of a Bayesian classifier.
Blayze - Blayze is a minimal JVM library for Naive Bayes classification written in Kotlin.
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values: {\displaystyle b+\mathbf {w} ^{\top }x>0}

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values: vertical-align: -0.838ex; width:59.545ex; height:3.176ex;

values: vertical-align: -0.838ex; width:16.147ex; height:2.843ex;

values: vertical-align: -0.838ex; margin-left: -0.089ex; width:33.753ex; height:3.176ex;

values: vertical-align: -0.838ex; margin-left: -0.089ex; width:34.138ex; height:3.176ex;

values: vertical-align: -0.838ex; margin-left: -0.089ex; width:35.571ex; height:3.176ex;

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values: vertical-align: -0.838ex; margin-left: -0.089ex; width:9.623ex; height:2.843ex;

values: vertical-align: -0.671ex; width:2.464ex; height:2.009ex;

values: {\displaystyle p(D\mid C)=\prod _{i}p(w_{i}\mid C)\,}

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values: vertical-align: -0.838ex; margin-left: -0.089ex; width:9.083ex; height:2.843ex;

values: {\displaystyle p(D\mid C)={p(D\cap C) \over p(C)}}

values: vertical-align: -2.671ex; margin-left: -0.089ex; width:21.882ex; height:6.509ex;

values: {\displaystyle p(C\mid D)={p(D\cap C) \over p(D)}}

values: {\displaystyle p(C\mid D)={\frac {p(C)\,p(D\mid C)}{p(D)}}}

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values: {\displaystyle p(D\mid S)=\prod _{i}p(w_{i}\mid S)\,}

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values: {\displaystyle p(D\mid \neg S)=\prod _{i}p(w_{i}\mid \neg S)\,}

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values:
/wiki/Log-likelihood_ratio
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Log-likelihood ratio

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Sigmoid curve

values: {\displaystyle p(S\mid D)>p(\neg S\mid D)}

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values: {\displaystyle \ln {p(S\mid D) \over p(\neg S\mid D)}>0}

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Bayesian spam filtering

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Random naive Bayes

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