ナイーブベイズ

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ナイーブベイズ#

すべての特徴量同士の関係について、目的変数を条件付けたもとでの条件付き独立をnaiveに仮定するモデル

特徴量x1,…,xnのもとでの目的変数yの確率P(y∣x1,…,xn)を次のように表す

P(y∣x1,…,xn)=P(y)P(x1,…,xn∣y)P(x1,…,xn)

条件付き独立の仮定により

P(xi|y,x1,…,xi−1,xi+1,…,xn)=P(xi|y)

なので式は簡素化され

P(y∣x1,…,xn)=P(y)∏i=1nP(xi∣y)P(x1,…,xn)

P(x1,…,xn)は入力を所与とすると定数なので

P(y∣x1,…,xn)∝P(y)∏i=1nP(xi∣y)⇓y^=arg⁡maxyP(y)∏i=1nP(xi∣y),

例#

線形分離不可能問題が解けない様子

# データの用意
import numpy as np
from sklearn.datasets import make_blobs
centers = [(1, 1), (1, -1), (-1, 1),  (-1, -1)]
X, y = make_blobs(n_samples=10000, n_features=2, centers=centers, cluster_std=[0.5, 0.5, 0.5, 0.5], random_state=0)

def replace_label(y):
    if y == 2:
        return 1
    if y == 3:
        return 0
    return y

y = np.array(list(map(replace_label, y)))

import matplotlib.pyplot as plt
fig, ax = plt.subplots()
for y_val in set(y):
    idx = y == y_val
    ax.scatter(X[idx, 0], X[idx, 1], label=f"y == {y_val}", alpha=0.3)
ax.legend()
fig.show()
../_images/ae06fd2b57903b3966e3f726a546c5615ddbf9d6fe5ed92e122515e71672ecdd.png 
from sklearn.model_selection import train_test_split
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33, random_state=42)

from sklearn.naive_bayes import BernoulliNB
clf = BernoulliNB()
clf.fit(X_train, y_train)

BernoulliNB()
In a Jupyter environment, please rerun this cell to show the HTML representation or trust the notebook.
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from sklearn.metrics import ConfusionMatrixDisplay
ConfusionMatrixDisplay.from_estimator(clf, X_test, y_test)

<sklearn.metrics._plot.confusion_matrix.ConfusionMatrixDisplay at 0x7fd6fc112a10>
../_images/40f7aad705a09d0fbcb33ccf8cb89b17fb493e2a1d1c398906b68edf4c5ad888.png
Hide code cell source 
from sklearn.inspection import DecisionBoundaryDisplay
fig, ax = plt.subplots()
disp = DecisionBoundaryDisplay.from_estimator(
    clf,
    X_test,
    response_method="predict",
    cmap=plt.cm.coolwarm,
    alpha=0.8,
    ax=ax,
    xlabel="x1",
    ylabel="x2",
)
ax.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.coolwarm, s=20, edgecolors="k")
ax.set(title="Decision Boundary of Naive Bayes")
fig.show()
../_images/b1b5a367e122f7ed0a26904e08c759846a8d98e04aab8f160edcd034c3b86a1a.png

参考文献#

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