Bayes classifiers

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Bayes classifiers

• Edgar Acuna

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Bayes Classifiers

• A formidable and sworn enemy of decision trees

BC
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How to build a Bayes Classifier

• Assume you want to predict output Y which has
  arity nY and values v1, v2, vny.
• Assume there are m input attributes called X1,
  X2, Xm
• Break dataset into nY smaller datasets called
  DS1, DS2, DSny.
• Define DSi Records in which Yvi
• For each DSi , learn Density Estimator Mi to
  model the input distribution among the Yvi
  records.

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How to build a Bayes Classifier

• Assume you want to predict output Y which has
  arity nY and values v1, v2, vny.
• Assume there are m input attributes called X1,
  X2, Xm
• Break dataset into nY smaller datasets called
  DS1, DS2, DSny.
• Define DSi Records in which Yvi
• For each DSi , learn Density Estimator Mi to
  model the input distribution among the Yvi
  records.
• Mi estimates P(X1, X2, Xm Yvi )

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How to build a Bayes Classifier

• Assume you want to predict output Y which has
  arity nY and values v1, v2, vny.
• Assume there are m input attributes called X1,
  X2, Xm
• Break dataset into nY smaller datasets called
  DS1, DS2, DSny.
• Define DSi Records in which Yvi
• For each DSi , learn Density Estimator Mi to
  model the input distribution among the Yvi
  records.
• Mi estimates P(X1, X2, Xm Yvi )
• Idea When a new set of input values (X1 u1, X2
  u2, . Xm um) come along to be evaluated
  predict the value of Y that makes P(X1, X2, Xm
  Yvi ) most likely

Is this a good idea?
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How to build a Bayes Classifier

• Assume you want to predict output Y which has
  arity nY and values v1, v2, vny.
• Assume there are m input attributes called X1,
  X2, Xm
• Break dataset into nY smaller datasets called
  DS1, DS2, DSny.
• Define DSi Records in which Yvi
• For each DSi , learn Density Estimator Mi to
  model the input distribution among the Yvi
  records.
• Mi estimates P(X1, X2, Xm Yvi )
• Idea When a new set of input values (X1 u1, X2
  u2, . Xm um) come along to be evaluated
  predict the value of Y that makes P(X1, X2, Xm
  Yvi ) most likely

This is a Maximum Likelihood classifier. It can
get silly if some Ys are very unlikely
Is this a good idea?
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How to build a Bayes Classifier

• Assume you want to predict output Y which has
  arity nY and values v1, v2, vny.
• Assume there are m input attributes called X1,
  X2, Xm
• Break dataset into nY smaller datasets called
  DS1, DS2, DSny.
• Define DSi Records in which Yvi
• For each DSi , learn Density Estimator Mi to
  model the input distribution among the Yvi
  records.
• Mi estimates P(X1, X2, Xm Yvi )
• Idea When a new set of input values (X1 u1, X2
  u2, . Xm um) come along to be evaluated
  predict the value of Y that makes P(Yvi X1,
  X2, Xm) most likely

Much Better Idea
Is this a good idea?
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Terminology

• MLE (Maximum Likelihood Estimator)

• MAP (Maximum A-Posteriori Estimator)

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Getting what we need
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Getting a posterior probability
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Bayes Classifiers in a nutshell
1. Learn the distribution over inputs for each
value Y. 2. This gives P(X1, X2, Xm Yvi
). 3. Estimate P(Yvi ). as fraction of records
with Yvi . 4. For a new prediction
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Bayes Classifiers in a nutshell
1. Learn the distribution over inputs for each
value Y. 2. This gives P(X1, X2, Xm Yvi
). 3. Estimate P(Yvi ). as fraction of records
with Yvi . 4. For a new prediction

• We can use our favorite Density Estimator here.
• Right now we have two options
• Joint Density Estimator
• Naïve Density Estimator

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Joint Density Bayes Classifier
In the case of the joint Bayes Classifier this
degenerates to a very simple rule Ypredict
the class containing most records in which X1
u1, X2 u2, . Xm um. Note that if no records
have the exact set of inputs X1 u1, X2 u2, .
Xm um, then P(X1, X2, Xm Yvi ) 0 for all
values of Y. In that case we just have to guess
Ys value
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Ejemplo
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Ejemplo Continuacion
X10,X20, x31 sera asignado a la clase 1 .
Notar tambien que en esta clase el record (0,0,1)
aparece mas veces que en la clase 0.
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Naïve Bayes Classifier
In the case of the naive Bayes Classifier this
can be simplified
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Naïve Bayes Classifier
In the case of the naive Bayes Classifier this
can be simplified
Technical Hint If you have 10,000 input
attributes that product will underflow in
floating point math. You should use logs
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Ejemplo Continuacion
X10,X20, x31 sera asignado a la clase 1
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BC Results XOR
The XOR dataset consists of 40,000 records and
2 Boolean inputs called a and b, generated 50-50
randomly as 0 or 1. c (output) a XOR b
The Classifier learned by Joint BC
The Classifier learned by Naive BC
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BC Results MPG 392 records
The Classifier learned by Naive BC
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More Facts About Bayes Classifiers

• Many other density estimators can be slotted in.
• Density estimation can be performed with
  real-valued inputs
• Bayes Classifiers can be built with real-valued
  inputs
• Rather Technical Complaint Bayes Classifiers
  dont try to be maximally discriminative---they
  merely try to honestly model whats going on
• Zero probabilities are painful for Joint and
  Naïve. A hack (justifiable with the magic words
  Dirichlet Prior) can help.
• Naïve Bayes is wonderfully cheap. And survives
  10,000 attributes cheerfully!
• See future Andrew Lectures

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Naïve Bayes classifier
Naïve Bayes classifier puede ser aplicado cuando
hay predictoras continuas, pero hay que aplicar
previamente un metodo de discretizacion tal
como Usando intervalos de igual ancho, usando
intervalos con igual frecuencia, ChiMerge,1R,
Discretizacion usando el metodo de la entropia
con distintos criterios de parada, Todos ellos
estan disponible en la libreria dprep( ver
disc.mentr, disc.ew, disc.ef, etc) . La libreria
e1071 de R contiene una funcion naiveBayes que
calcula el clasificador naïve Bayes. Si la
variable es continua asume que sigue una
distribucion Gaussiana.
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The misclassification error rate
The misclassification error rate R(d) is the
probability that the classifier d classifies
incorrectly an instance coming from a sample
(test sample) obtained in a later stage than the
training sample. Also is called the True error or
the actual error. It is an unknown value that
needs to be estimated.
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Methods for estimation of the misclassification
error rate

• Resubstitution or Aparent Error (Smith, 1947).
  This is merely the proportion of instances in the
  training sample that are incorrectly classified
  by the classification rule. In general is an
  estimator too optimistic and it can lead to wrong
  conclusions if the number of instances is not
  large compared with the number of features. This
  estimator has a large bias.
• ii) Leave one out estimation. (Lachenbruch,
  1965). In this case an instance is omitted from
  the training sample. Then the classifier is built
  and the prediction for the omitted instances is
  obtained. One must register if the instance was
  correctly or incorrectly classfied. The process
  is repeated for all the instances in the training
  sample and the estimation of the ME will be given
  by the proportion of instances incorrectly
  classified. This estimator has low bias but its
  variance tends to be large.

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Methods for estimation of the misclassification
error rate

• iii) Cross validation. (Stone, 1974) In this case
  the training sample is randomly divided in v
  parts (v10 is the most used). Then the
  classifier is built using all the parts but one.
  The omitted part is considered as the test sample
  and the predictions for each instance on it are
  found. The CV misclassification error rate is
  found by adding the misclassification on each
  part and dividing them by the total number of
  instances. The CV estimated has low bias but
  high variance. In order to reduce the variability
  we usually repeat the estimation several times.
• The estimation of the variance is a hard problem
  (bengio and Grandvalet, 2004).

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Methods for estimation of the misclassification
error rate
iv) The holdout method. A percentage (70) of the
dataset is considered as the training sample and
the remaining as the test sample. The classifier
is evaluated in the test sample. The experiment
is repeated several times and then the average is
taken. v) Bootstrapping. (Efron, 1983). In this
method we generate several training samples by
sampling with replacement from the original
training sample. The idea is to reduce the bias
of the resubstitution error. It is almost
unbiased, but it has a large variance. Its
computation cost is high. There exist several
variants of this method.
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Naive Bayes para Bupa
Sin discretizar gt anaiveBayes(V7.,databupa) gt
predpredict(a,bupa,-7,type"raw") gt
pred1max.col(pred) gt table(pred1,bupa,7)
pred1 1 2 1 112 119 2 33 81 gt
error152/345 1 0.4405797 Discretizando con el
metodo de la entropia gt dbupadisc.mentr(bupa,17
) gt bnaiveBayes(V7.,datadbupa) gt
predpredict(b,dbupa,-7) gt table(pred,dbupa,7)
pred 1 2 1 79 61 2 66 139 gt
error1127/345 1 0.3681159
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Naïve Bayes para Diabetes
Sin Descritizar gt anaiveBayes(V9.,datadiabetes)
gt predpredict(a,diabetes,-9,type"raw") gt
pred1max.col(pred) gt table(pred1,diabetes,9)
pred1 1 2 1 421 104 2 79 164 gt
error(79104)/768 1 0.2382813 Discretizando gt
ddiabetesdisc.mentr(diabetes,19) gt
bnaiveBayes(V9.,dataddiabetes) gt
predpredict(b,ddiabetes,-9) gt
table(pred,ddiabetes,9) pred 1 2 1
418 84 2 82 184 gt 166/768 1 0.2161458
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Naïve Bayes usando discretizacion ChiMerge
gt chibupachiMerge(bupa,16) gt bnaiveBayes(V7.,d
atachibupa) gt predpredict(b,chibupa,-7) gt
table(pred,chibupa,7) pred 1 2 1 117
21 2 28 179 gt error49/345 1 0.1420290 gt
chidiabchiMerge(diabetes,18) gt
bnaiveBayes(V9.,datachidiab) gt
predpredict(b,chidiab,-9) gt table(pred,chidiab
,9) pred 1 2 1 457 33 2 43 235 gt
error76/768 1 0.09895833
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Otros clasificadores Bayesianos
Analisis Discriminante Lineal (LDA). Aqui se
asume que la funcion de clase condicional
P(X1,Xm/Yvj) se asume que es normal
multivariada para cada vj. Se supone ademas que
la matriz de covarianza es igual para cada una de
las clases. La regla de decision para asignar el
objeto x se reduce a Notar que la regla de
decision es lineal en el vector de predictoras x.
Estrictamente hablando solo deberia aplicarse
cuando las predictoras son continuas.
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Ejemplos de LDABupa y Diabetes
gt bupaldalda(V7.,databupa) gt
predpredict(bupalda,bupa,-7)class gt
table(pred,bupa,7) pred 1 2 1 78
35 2 67 165 gt error102/345 1 0.2956522 gt
diabetesldalda(V9.,datadiabetes) gt
predpredict(diabeteslda,diabetes,-9)class gt
table(pred,diabetes,9) pred 1 2 1
446 112 2 54 156 gt error166/768 1 0.2161458
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Otros clasificadores Bayesianos
Los k vecinos mas cercanos (k nearest
neighbor). Aqui la funcion de clase condicional
P(X1,Xm/Yvj) es estimada por el metodo de los
k-vecinos mas cercanos. Estimadores basados en
estimacion de densidad por Kernel. Estimadores
basados en estimacion de la densidad condiiconal
usando mezclas Gaussianas
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El clasificador k-nn

• En el caso multivariado, el estimado de la
  función de densidad tiene la forma
• donde vk(x) es el volumen de un elipsoide
  centrado en x de radio rk(x), que a su vez es la
  distancia de x al k-ésimo punto más cercano.

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El clasificador k-nn
Desde el punto de vista de clasificacion
supervisada el método k-nn es bien simple de
aplicar. En efecto, si las funciones de
densidades condicionales f(x/Ci) de la clase Ci
que aparecen en la ecuación son estimadas por
k-nn. Entonces, para clasificar un objeto, con
mediciones dadas por el vector x, en la clase Ci
se debe cumplir que para j?i. Donde ki y kj
son los k vecinos de x que caen en las clase Ci y
Cj respectivamente.
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El clasificador k-nn
Asumiendo priors proporcionales a los tamaños de
las clases (ni/n y nj/n respectivamente) lo
anterior es equivalente a
kigtkj para j?iLuego, el procedimiento de
clasificación sería así 1) Hallar los k objetos
que están a una distancia más cercana al ojbeto
x, k usualmente es un número impar 1 o 3. 2) Si
la mayoría de esos k objetos pertenecen a la
clase Ci entonces el objeto x es asignado a
ella. En caso de empate se clasifica al azar.
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El clasificador k-nn

• Hay dos problemas en el método k-nn, la elección
  de la distancia o métrica y la elección de k.
• La métrica más elemental que se puede elegir es
  la euclideana d(x,y)(x-y)'(x-y). Esta métrica
  sin embargo, puede causar problemas si las
  variables predictoras han sido medidas en
  unidades muy distintas entre sí. Algunos
  prefieren rescalar los datos antes de aplicar el
  método. Otra distancia bien usada es la distancia
  Manhatan definida por d(x,y)x-y. Hay metricas
  especiales cuando hay distintode variables en el
  conjunto de datos.
• Enas y Choi (1996) usando simulación hicieron un
  estudio para determinar el k óptimo cuando solo
  hay dos clases presentesy determinaron que si los
  tamaños muestrales de las dos clases son
  comparables entonces kn3/8 si habia poca
  diferencia entre las matrices de covarianzas de
  los grupos y kn2/8 si habia bastante diferencia
  entre las matrices de covarianzas.

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Ejemplo de knn Bupa
gt bupak1knn(bupa,-7,bupa,-7,as.factor(bupa,7
),k1) gt table(bupak1,bupa,7) bupak1 1 2
1 145 0 2 0 200 error0 gt
bupak3knn(bupa,-7,bupa,-7,as.factor(bupa,7)
,k3) gt table(bupak3,bupa,7) bupak3 1
2 1 106 29 2 39
171 error19.71 gt bupak5knn(bupa,-7,bupa,-7,
as.factor(bupa,7),k5) gt table(bupak5,bupa,7)
bupak5 1 2 1 94 23 2 51
177 error21.44
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Ejemplo de knn diabetes
gt diabk3knn(diabetes,-9,diabetes,-9,as.factor
(diabetes,9),k3) gt table(diabk3,diabetes,9)
diabk1 1 2 1 459 67 2 41
201 error14.06 gt diabk5knn(diabetes,-9,diabet
es,-9,as.factor(diabetes,9),k5) gt
table(diabk5,diabetes,9) diabk1 1 2
1 442 93 2 58 175 error19.66
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What you should know

• Bayes Classifiers
• How to build one
• How to predict with a BC
• How to estimate the misclassification error.