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\(x_1,\dots,x_n\)のもとでの目的変数\(y\)の確率\(P(y\mid x_1,\dots, x_n)\)を次のように表す
\[
P\left(y \mid x_1, \ldots, x_n\right)=\frac{P(y) P\left(x_1, \ldots, x_n \mid y\right)}{P\left(x_1, \ldots, x_n\right)}
\]
条件付き独立の仮定により
\[
P(x_i | y, x_1, \dots, x_{i-1}, x_{i+1}, \dots, x_n) = P(x_i | y)
\]
なので式は簡素化され
\[
P(y \mid x_1, \dots, x_n) = \frac{P(y) \prod_{i=1}^{n} P(x_i \mid y)}
{P(x_1, \dots, x_n)}
\]
\(P(x_1, \dots, x_n)\)は入力を所与とすると定数なので
\[\begin{split}
\begin{align}\begin{aligned}P(y \mid x_1, \dots, x_n) \propto P(y) \prod_{i=1}^{n} P(x_i \mid y)\\\Downarrow\\\hat{y} = \arg\max_y P(y) \prod_{i=1}^{n} P(x_i \mid y),\end{aligned}\end{align}
\end{split}\]
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()
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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BernoulliNB()
from sklearn.metrics import ConfusionMatrixDisplay
ConfusionMatrixDisplay.from_estimator(clf, X_test, y_test)
<sklearn.metrics._plot.confusion_matrix.ConfusionMatrixDisplay at 0x7ff235bb9ea0>
Show 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()
