Contents

Warning: This MADlib method is still in early stage development. Interface and implementation are subject to change.

Naive Bayes refers to a stochastic model where all independent variables \( a_1, \dots, a_n \) (often referred to as attributes in this context) independently contribute to the probability that a data point belongs to a certain class \( c \).

Naives Bayes classification estimates feature probabilities and class priors using maximum likelihood or Laplacian smoothing. For numeric attributes, Gaussian smoothing can be used to estimate the feature probabilities.These parameters are then used to classify new data.

Training Function(s)

For data with only categorical attributes, precompute feature probabilities and class priors using the following function:

create_nb_prepared_data_tables ( trainingSource,
                                 trainingClassColumn,
                                 trainingAttrColumn,
                                 numAttrs,
                                 featureProbsName,
                                 classPriorsName
                               )

For data containing both categorical and numeric attributes, use the following form to precompute the Gaussian parameters (mean and variance) for numeric attributes alongside the feature probabilities for categorical attributes and class priors.

create_nb_prepared_data_tables ( trainingSource,
                                 trainingClassColumn,
                                 trainingAttrColumn,
                                 numericAttrsColumnIndices,
                                 numAttrs,
                                 featureProbsName,
                                 numericAttrParamsName,
                                 classPriorsName
                               )

The trainingSource is expected to be of the following form:

{TABLE|VIEW} trainingSource (
    ...
    trainingClassColumn INTEGER,
    trainingAttrColumn INTEGER[] OR NUMERIC[] OR FLOAT8[],
    ...
)

numericAttrsColumnIndices should be of type TEXT, specified as an array of indices (starting from 1) in the trainingAttrColumn attributes-array that correspond to numeric attributes.

The two output tables are:

featureProbsName – stores feature probabilities
classPriorsName – stores the class priors

In addition to the above, if the function specifying numeric attributes is used, an additional table numericAttrParamsName is created which stores the Gaussian parameters for the numeric attributes.

Classify Function(s)

Perform Naive Bayes classification:

create_nb_classify_view ( featureProbsName,
                          classPriorsName,
                          classifySource,
                          classifyKeyColumn,
                          classifyAttrColumn,
                          numAttrs,
                          destName
                        )

For data with numeric attributes, use the following version:

create_nb_classify_view ( featureProbsName,
                          classPriorsName,
                          classifySource,
                          classifyKeyColumn,
                          classifyAttrColumn,
                          numAttrs,
                          numericAttrParamsName,
                          destName
                        )

The data to classify is expected to be of the following form:

{TABLE|VIEW} classifySource (
    ...
    classifyKeyColumn ANYTYPE,
    classifyAttrColumn INTEGER[],
    ...
)

This function creates the view destName mapping classifyKeyColumn to the Naive Bayes classification.

key | nb_classification
 ---+------------------
...

Probabilities Function(s)

Compute Naive Bayes probabilities.

create_nb_probs_view( featureProbsName,
                      classPriorsName,
                      classifySource,
                      classifyKeyColumn,
                      classifyAttrColumn,
                      numAttrs,
                      destName
                    )

For data with numeric attributes , use the following version:

create_nb_probs_view( featureProbsName,
                      classPriorsName,
                      classifySource,
                      classifyKeyColumn,
                      classifyAttrColumn,
                      numAttrs,
                      numericAttrParamsName,
                      destName
                    )

This creates the view destName mapping classifyKeyColumn and every single class to the Naive Bayes probability:

key | class | nb_prob
 ---+-------+--------
...

Ad Hoc Computation Function

With ad hoc execution (no precomputation), the functions create_nb_classify_view() and create_nb_probs_view() can be used in an ad-hoc fashion without the precomputation step. In this case, replace the function arguments

'featureProbsName', 'classPriorsName'

with

'trainingSource', 'trainingClassColumn', 'trainingAttrColumn'

for data without any any numeric attributes and with

'trainingSource', 'trainingClassColumn', 'trainingAttrColumn', 'numericAttrsColumnIndices'

for data containing numeric attributes as well.

Implementation Notes

The probabilities computed on the platforms of PostgreSQL and Greenplum database have a small difference due to the nature of floating point computation. Usually this is not important. However, if a data point has
\[ P(C=c_i \mid A) \approx P(C=c_j \mid A) \]
for two classes, this data point might be classified into diferent classes on PostgreSQL and Greenplum. This leads to the differences in classifications on PostgreSQL and Greenplum for some data sets, but this should not affect the quality of the results.
When two classes have equal and highest probability among all classes, the classification result is an array of these two classes, but the order of the two classes is random.
The current implementation of Naive Bayes classification is suitable for discontinuous (categorial) attributes as well as continuous (numeric) attributes.
For continuous data, a typical assumption, usually used for small datasets, is that the continuous values associated with each class are distributed according to a Gaussian distribution, and the probabilities \( P(A_i = a \mid C=c) \) are estimated using the Gaussian Distribution formula:
\[ P(A_i=a \mid C=c) = \frac{1}{\sqrt{2\pi\sigma^{2}_c}}exp\left(-\frac{(a-\mu_c)^{2}}{2\sigma^{2}_c}\right) \]
where \(\mu_c\) and \(\sigma^{2}_c\) are the population mean and variance of the attribute for the class \(c\).
Another common technique for handling continuous values, which is better for large data sets, is to use binning to discretize the values, and convert the continuous data into categorical bins. This approach is currently not implemented.
One can provide floating point data to the Naive Bayes classification function. If the corresponding attribute index is not specified in numericAttrsColumnIndices, floating point numbers will be used as symbolic substitutions for categorial data. In this case, the classification would work best if there are sufficient data points for each floating point attribute. However, if floating point numbers are used as continuous data without the attribute being marked as of type numeric in numericAttrsColumnIndices, no warning is raised and the result may not be as expected.

Examples

The following is an extremely simplified example of the above option #1 which can by verified by hand.

The training and the classification data.

SELECT * FROM training;

Result:

 id | class | attributes
 ---+-------+------------
  1 |     1 | {1,2,3}
  2 |     1 | {1,2,1}
  3 |     1 | {1,4,3}
  4 |     2 | {1,2,2}
  5 |     2 | {0,2,2}
  6 |     2 | {0,1,3}
(6 rows)

SELECT * FROM toclassify;

Result:

 id | attributes
 ---+------------
  1 | {0,2,1}
  2 | {1,2,3}
(2 rows)

Precompute feature probabilities and class priors.

SELECT madlib.create_nb_prepared_data_tables( 'training',
                                              'class',
                                              'attributes',
                                              3,
                                              'nb_feature_probs',
                                              'nb_class_priors'
                                            );

Optionally check the contents of the precomputed tables.

SELECT * FROM nb_class_priors;

Result:

 class | class_cnt | all_cnt
 ------+-----------+---------
     1 |         3 |       6
     2 |         3 |       6
(2 rows)

SELECT * FROM nb_feature_probs;

Result:

 class | attr | value | cnt | attr_cnt
 ------+------+-------+-----+----------
     1 |    1 |     0 |   0 |        2
     1 |    1 |     1 |   3 |        2
     1 |    2 |     1 |   0 |        3
     1 |    2 |     2 |   2 |        3
...

Create the view with Naive Bayes classification and check the results.

SELECT madlib.create_nb_classify_view( 'nb_feature_probs',
                                       'nb_class_priors',
                                       'toclassify',
                                       'id',
                                       'attributes',
                                       3,
                                       'nb_classify_view_fast'
                                     );
 
SELECT * FROM nb_classify_view_fast;

Result:

 key | nb_classification
 ----+-------------------
   1 | {2}
   2 | {1}
(2 rows)

Look at the probabilities for each class (note that we use "Laplacian smoothing"),

SELECT madlib.create_nb_probs_view( 'nb_feature_probs',
                                    'nb_class_priors',
                                    'toclassify',
                                    'id',
                                    'attributes',
                                    3,
                                    'nb_probs_view_fast'
                                  );
 
SELECT * FROM nb_probs_view_fast;

Result:

 key | class | nb_prob
 ----+-------+---------
   1 |     1 |     0.4
   1 |     2 |     0.6
   2 |     1 |    0.75
   2 |     2 |    0.25
(4 rows)

The following is an example of using a dataset with both numeric and categorical attributes

The training and the classification data. Attributes {height(numeric),weight(numeric),shoe size(categorical)}, Class{sex(1=male,2=female)}

SELECT * FROM gaussian_data;

Result:

 id | sex |  attributes
 ----+-----+---------------
  1 |   1 | {6,180,12}
  2 |   1 | {5.92,190,12}
  3 |   1 | {5.58,170,11}
  4 |   1 | {5.92,165,11}
  5 |   2 | {5,100,6}
  6 |   2 | {5.5,150,6}
  7 |   2 | {5.42,130,7}
  8 |   2 | {5.75,150,8}
(8 rows)

SELECT * FROM gaussian_test;

Result:

 id | sex |  attributes
----+-----+--------------
  9 |   1 | {5.8,180,11}
 10 |   2 | {5,160,6}
(2 rows)

Precompute feature probabilities and class priors.

SELECT madlib.create_nb_prepared_data_tables( 'gaussian_data',
                                              'sex',
                                              'attributes',
                                              'ARRAY[1,2]',
                                              3,
                                              'categ_feature_probs',
                                              'numeric_attr_params',
                                              'class_priors'
                                            );

Optionally check the contents of the precomputed tables.

SELECT * FROM class_priors;

Result:

class | class_cnt | all_cnt
 -------+-----------+---------
     1 |         4 |       8
     2 |         4 |       8
(2 rows)

SELECT * FROM categ_feature_probs;

Result:

 class | attr | value | cnt | attr_cnt
-------+------+-------+-----+----------
     2 |    3 |     6 |   2 |        5
     1 |    3 |    12 |   2 |        5
     2 |    3 |     7 |   1 |        5
     1 |    3 |    11 |   2 |        5
     2 |    3 |     8 |   1 |        5
     2 |    3 |    12 |   0 |        5
     1 |    3 |     6 |   0 |        5
     2 |    3 |    11 |   0 |        5
     1 |    3 |     8 |   0 |        5
     1 |    3 |     7 |   0 |        5
(10 rows)

SELECT * FROM numeric_attr_params;

Result:

class | attr |      attr_mean       |        attr_var
-------+------+----------------------+------------------------
     1 |    1 |   5.8550000000000000 | 0.03503333333333333333
     1 |    2 | 176.2500000000000000 |   122.9166666666666667
     2 |    1 |   5.4175000000000000 | 0.09722500000000000000
     2 |    2 | 132.5000000000000000 |   558.3333333333333333
(4 rows)

Create the view with Naive Bayes classification and check the results.

SELECT madlib.create_nb_classify_view( 'categ_feature_probs',
                                       'class_priors',
                                       'gaussian_test',
                                       'id',
                                       'attributes',
                                       3,
                                       'numeric_attr_params',
                                       'classify_view'
                                     );
 
SELECT * FROM classify_view;

Result:

 key | nb_classification
 ----+-------------------
   9 | {1}
   10 | {2}
(2 rows)

Look at the probabilities for each class

SELECT madlib.create_nb_probs_view( 'categ_feature_probs',
                                       'class_priors',
                                       'gaussian_test',
                                       'id',
                                       'attributes',
                                       3,
                                       'numeric_attr_params',
                                       'probs_view'
                                  );
 
SELECT * FROM probs_view;

Result:

 key | class |       nb_prob
-----+-------+----------------------
   9 |     1 |    0.993556745948775
   9 |     2 |  0.00644325405122553
  10 |     1 | 5.74057538627122e-05
  10 |     2 |    0.999942594246137
(4 rows)

Technical Background

In detail, Bayes' theorem states that

\[ \Pr(C = c \mid A_1 = a_1, \dots, A_n = a_n) = \frac{\Pr(C = c) \cdot \Pr(A_1 = a_1, \dots, A_n = a_n \mid C = c)} {\Pr(A_1 = a_1, \dots, A_n = a_n)} \,, \]

and the naive assumption is that

\[ \Pr(A_1 = a_1, \dots, A_n = a_n \mid C = c) = \prod_{i=1}^n \Pr(A_i = a_i \mid C = c) \,. \]

Naives Bayes classification estimates feature probabilities and class priors using maximum likelihood or Laplacian smoothing. These parameters are then used to classifying new data.

A Naive Bayes classifier computes the following formula:

\[ \text{classify}(a_1, ..., a_n) = \arg\max_c \left\{ \Pr(C = c) \cdot \prod_{i=1}^n \Pr(A_i = a_i \mid C = c) \right\} \]

where \( c \) ranges over all classes in the training data and probabilites are estimated with relative frequencies from the training set. There are different ways to estimate the feature probabilities \( P(A_i = a \mid C = c) \). The maximum likelihood estimate takes the relative frequencies. That is:

\[ P(A_i = a \mid C = c) = \frac{\#(c,i,a)}{\#c} \]

where

\( \#(c,i,a) \) denotes the # of training samples where attribute \( i \) is \( a \) and class is \( c \)
\( \#c \) denotes the # of training samples where class is \( c \).

Since the maximum likelihood sometimes results in estimates of "0", you might want to use a "smoothed" estimate. To do this, you add a number of "virtual" samples and make the assumption that these samples are evenly distributed among the values assumed by attribute \( i \) (that is, the set of all values observed for attribute \( a \) for any class):

\[ P(A_i = a \mid C = c) = \frac{\#(c,i,a) + s}{\#c + s \cdot \#i} \]

where

\( \#i \) denotes the # of distinct values for attribute \( i \) (for all classes)
\( s \geq 0 \) denotes the smoothing factor.

The case \( s = 1 \) is known as "Laplace smoothing". The case \( s = 0 \) trivially reduces to maximum-likelihood estimates.

Literature

[1] Tom Mitchell: Machine Learning, McGraw Hill, 1997. Book chapter Generativ and Discriminative Classifiers: Naive Bayes and Logistic Regression available at: http://www.cs.cmu.edu/~tom/NewChapters.html

[2] Wikipedia, Naive Bayes classifier, http://en.wikipedia.org/wiki/Naive_Bayes_classifier

Related Topics: File bayes.sql_in documenting the SQL functions.