@@ -224,6 +224,40 @@ It is advisable to evaluate both models, if time permits.
224224 <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.61.5542> `_
225225 3rd Conf. on Email and Anti-Spam (CEAS).
226226
227+ .. _categorical_naive_bayes :
228+
229+ Categorical Naive Bayes
230+ -----------------------
231+
232+ :class: `CategoricalNB ` implements the categorical naive Bayes
233+ algorithm for categorically distributed data. It assumes that each feature,
234+ which is described by the index :math: `i`, has its own categorical
235+ distribution.
236+
237+ For each feature :math: `i` in the training set :math: `X`,
238+ :class: `CategoricalNB ` estimates a categorical distribution for each feature i
239+ of X conditioned on the class y. The index set of the samples is defined as
240+ :math: `J = \{ 1 , \dots , m \}`, with :math: `m` as the number of samples.
241+
242+ The probability of category :math: `t` in feature :math: `i` given class
243+ :math: `c` is estimated as:
244+
245+ .. math ::
246+
247+ P(x_i = t \mid y = c \: ;\, \alpha ) = \frac { N_{tic} + \alpha }{N_{c} +
248+ \alpha n_i},
249+
250+ :math: `N_{tic} = |\{ j \in J \mid x_{ij} = t, y_j = c\}|` is the number
251+ of times category :math: `t` appears in the samples :math: `x_{i}`, which belong
252+ to class :math: `c`, :math: `N_{c} = |\{ j \in J\mid y_j = c\}|` is the number
253+ of samples with class c, :math: `\alpha ` is a smoothing parameter and
254+ :math: `n_i` is the number of available categories of feature :math: `i`.
255+
256+ :class: `CategoricalNB ` assumes that the sample matrix :math: `X` is encoded
257+ (for instance with the help of :class: `OrdinalEncoder `) such that all
258+ categories for each feature :math: `i` are represented with numbers
259+ :math: `0 , ..., n_i - 1 ` where :math: `n_i` is the number of available categories
260+ of feature :math: `i`.
227261
228262Out-of-core naive Bayes model fitting
229263-------------------------------------
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