# 1.9. ナイーブベイズ

ナイーブベイズは、ベイズの定理をベースにして、全ての特徴量のペア間で独立した”単純な”推定をする、教師あり学習アルゴリズムの一セットです。分類ラベル と依存している特徴量ベクトル から が与えられるとベイズの定理の状態は次のような関係になります。

Using the naive independence assumption that

P(x_i | y, x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_n) = P(x_i | y), for all i, this relationship is simplified to P(y \mid x_1, \dots, x_n) = \frac{P(y) \prod_{i=1}^{n} P(x_i \mid y)} {P(x_1, \dots, x_n)} Since P(x_1, \dots, x_n) is constant given the input, we can use the following classification rule: P(y \mid x_1, \dots, x_n) \propto P(y) \prod_{i=1}^{n} P(x_i \mid y)

\Downarrow

\hat{y} = \arg\max_y P(y) \prod_{i=1}^{n} P(x_i \mid y), and we can use Maximum A Posteriori (MAP) estimation to estimate P(y) and P(x_i \mid y); the former is then the relative frequency of class y in the training set. The different naive Bayes classifiers differ mainly by the assumptions they make regarding the distribution of P(x_i \mid y). In spite of their apparently over-simplified assumptions, naive Bayes classifiers have worked quite well in many real-world situations, famously document classification and spam filtering. They require a small amount of training data to estimate the necessary parameters. (For theoretical reasons why naive Bayes works well, and on which types of data it does, see the references below.) Naive Bayes learners and classifiers can be extremely fast compared to more sophisticated methods. The decoupling of the class conditional feature distributions means that each distribution can be independently estimated as a one dimensional distribution. This in turn helps to alleviate problems stemming from the curse of dimensionality. On the flip side, although naive Bayes is known as a decent classifier, it is known to be a bad estimator, so the probability outputs from predict_proba are not to be taken too seriously. References: H. Zhang (2004). The optimality of Naive Bayes. Proc. FLAIRS.

## 1.9.1. ガウス分布ナイーブベイズ

GaussianNBはガウス分布ナイーブベイスの分類アルゴリズムを実装します。

特徴の見込みは、ガウス分布で想定されます:

パラメータの と は最大の可能性のもので推定されます。

```
>>> from sklearn import datasets
>>> iris = datasets.load_iris()
>>> from sklearn.naive_bayes import GaussianNB
>>> gnb = GaussianNB()
>>> y_pred = gnb.fit(iris.data, iris.target).predict(iris.data)
>>> print("Number of mislabeled points out of a total %d points : %d"
... % (iris.data.shape[0],(iris.target != y_pred).sum()))
Number of mislabeled points out of a total 150 points : 6
```

## 1.9.2. Multinomial Naive Bayes

MultinomialNB implements the naive Bayes algorithm for multinomially distributed data, and is one of the two classic naive Bayes variants used in text classification (where the data are typically represented as word vector counts, although tf-idf vectors are also known to work well in practice). The distribution is parametrized by vectors \theta_y = (\theta_{y1},\ldots,\theta_{yn}) for each class y, where n is the number of features (in text classification, the size of the vocabulary) and \theta_{yi} is the probability P(x_i \mid y) of feature i appearing in a sample belonging to class y.
The parameters \theta_y is estimated by a smoothed version of maximum likelihood, i.e. relative frequency counting:
\hat{\theta}*{yi} = \frac{ N*{yi} + \alpha}{N_y + \alpha n}
where N_{yi} = \sum_{x \in T} x_i is the number of times feature i appears in a sample of class y in the training set T, and N_{y} = \sum_{i=1}^{|T|} N_{yi} is the total count of all features for class y.
The smoothing priors \alpha \ge 0 accounts for features not present in the learning samples and prevents zero probabilities in further computations. Setting \alpha = 1 is called Laplace smoothing, while \alpha < 1 is called Lidstone smoothing.

## 1.9.3. Bernoulli Naive Bayes

BernoulliNB implements the naive Bayes training and classification algorithms for data that is distributed according to multivariate Bernoulli distributions; i.e., there may be multiple features but each one is assumed to be a binary-valued (Bernoulli, boolean) variable. Therefore, this class requires samples to be represented as binary-valued feature vectors; if handed any other kind of data, a BernoulliNB instance may binarize its input (depending on the binarize parameter). The decision rule for Bernoulli naive Bayes is based on P(x_i \mid y) = P(i \mid y) x_i + (1 - P(i \mid y)) (1 - x_i) which differs from multinomial NB’s rule in that it explicitly penalizes the non-occurrence of a feature i that is an indicator for class y, where the multinomial variant would simply ignore a non-occurring feature. In the case of text classification, word occurrence vectors (rather than word count vectors) may be used to train and use this classifier. BernoulliNB might perform better on some datasets, especially those with shorter documents. It is advisable to evaluate both models, if time permits. References: C.D. Manning, P. Raghavan and H. Schütze (2008). Introduction to Information Retrieval. Cambridge University Press, pp. 234-265. A. McCallum and K. Nigam (1998). A comparison of event models for Naive Bayes text classification. Proc. AAAI/ICML-98 Workshop on Learning for Text Categorization, pp. 41-48. V. Metsis, I. Androutsopoulos and G. Paliouras (2006). Spam filtering with Naive Bayes – Which Naive Bayes? 3rd Conf. on Email and Anti-Spam (CEAS). 1.9.4. Out-of-core naive Bayes model fitting

Naive Bayes models can be used to tackle large scale classification problems for which the full training set might not fit in memory. To handle this case, MultinomialNB, BernoulliNB, and GaussianNB expose a partial_fit method that can be used incrementally as done with other classifiers as demonstrated in Out-of-core classification of text documents. All naive Bayes classifiers support sample weighting. Contrary to the fit method, the first call to partial_fit needs to be passed the list of all the expected class labels. For an overview of available strategies in scikit-learn, see also the out-of-core learning documentation. Note The partial_fit method call of naive Bayes models introduces some computational overhead. It is recommended to use data chunk sizes that are as large as possible, that is as the available RAM allows.

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