Machine Learning 简明教程

Machine Learning - Bayes Theorem

贝叶斯定理是概率论中的一个基本概念，在机器学习中有很多应用。它允许我们根据新证据更新我们对事件概率的信念。实际上，它形成了概率推理和决策的基矗。

Bayes Theorem is a fundamental concept in probability theory that has many applications in machine learning. It allows us to update our beliefs about the probability of an event given new evidence. Actually, it forms the basis for probabilistic reasoning and decision making.

贝叶斯定理指出，给定证据 B 的事件 A 的概率等于给定事件 A 的证据 B 的概率乘以事件 A 的先验概率再除以证据 B 的概率。在数学符号中，可以写成 −

Bayes Theorem states that the probability of an event A given evidence B is equal to the probability of evidence B given event A, multiplied by the prior probability of event A, divided by the probability of evidence B. In mathematical notation, this can be written as −

P\left ( A| B \right )=P\left ( B|A \right )\ast P\left ( A \right )/P\left ( B \right )

其中：

where −

  1. $P\left ( A| B \right )$ is the probability of event A given evidence B (the posterior probability)

  2. $P\left ( B|A \right )$ is the probability of evidence B given event A (the likelihood)

  3. $P\left ( A \right )$ is the prior probability of event A (our initial belief about the probability of event A)

  4. $P\left ( B \right )$ is the probability of evidence B (the total probability)

贝叶斯定理可以在广泛的应用中使用，例如垃圾邮件过滤、医疗诊断和图像识别。在机器学习中，贝叶斯定理通常用于贝叶斯推断，这是一种基于新数据更新我们对模型参数的信念的统计技术。

Bayes Theorem can be used in a wide range of applications, such as spam filtering, medical diagnosis, and image recognition. In machine learning, Bayes Theorem is commonly used in Bayesian inference, which is a statistical technique for updating our beliefs about the parameters of a model based on new data.

Implementation in Python

在 Python 中，有几个库实现了贝叶斯定理和贝叶斯推断。最受欢迎的一个是 scikit-learn 库，它为机器学习和数据分析提供了一系列工具。

In Python, there are several libraries that implement Bayes Theorem and Bayesian inference. One of the most popular is the scikit-learn library, which provides a range of tools for machine learning and data analysis.

我们来看一个使用 scikit-learn 在 Python 中实现贝叶斯定理的示例。假设我们有一个电子邮件数据集，其中一些是垃圾邮件，一些不是。我们的目标是构建一个能够准确预测新邮件是否是垃圾邮件的分类器。

Let’s consider an example of how Bayes Theorem can be implemented in Python using scikit-learn. Suppose we have a dataset of emails, some of which are spam and some of which are not. Our goal is to build a classifier that can accurately predict whether a new email is spam or not.

我们可以使用贝叶斯定理计算给定其特征（例如主题行或正文中的单词）的情况下电子邮件是垃圾邮件的概率。为此，我们首先需要估计模型的参数，在本例中，这些参数是垃圾邮件和非垃圾邮件的先验概率，以及给定类（垃圾邮件或非垃圾邮件）情况下每个特征的可能性。

We can use Bayes Theorem to calculate the probability of an email being spam given its features (such as the words in the subject line or body). To do this, we first need to estimate the parameters of the model, which in this case are the prior probabilities of spam and non-spam emails, as well as the likelihood of each feature given the class (spam or non-spam).

我们可以使用极大似然估计或贝叶斯推断来估计这些概率。在我们的示例中，我们将使用多项式朴素贝叶斯算法，这是一种朴素贝叶斯算法的变体，通常用于文本分类任务。

We can estimate these probabilities using maximum likelihood estimation or Bayesian inference. In our example, we will be using the Multinomial Naive Bayes algorithm, which is a variant of the Naive Bayes algorithm that is commonly used for text classification tasks.

Example

from sklearn.datasets import fetch_20newsgroups
from sklearn.feature_extraction.text import CountVectorizer
from sklearn.naive_bayes import MultinomialNB
from sklearn.metrics import accuracy_score

# Load the 20 newsgroups dataset
categories = ['alt.atheism', 'comp.graphics', 'sci.med', 'soc.religion.christian']
train = fetch_20newsgroups(subset='train', categories=categories, shuffle=True, random_state=42)
test = fetch_20newsgroups(subset='test', categories=categories, shuffle=True, random_state=42)

# Vectorize the text data using a bag-of-words representation
vectorizer = CountVectorizer()
X_train = vectorizer.fit_transform(train.data)
X_test = vectorizer.transform(test.data)

# Train a Multinomial Naive Bayes classifier
clf = MultinomialNB()
clf.fit(X_train, train.target)

# Make predictions on the test set and calculate accuracy
y_pred = clf.predict(X_test)
accuracy = accuracy_score(test.target, y_pred)
print("Accuracy:", accuracy)

在上面的代码中，我们首先加载 20 个新闻组数据集，这是按不同类别分类的一系列新闻组帖子。我们选择四种类别（alt.atheism、comp.graphics、sci.med 和 soc.religion.christian），并将数据分成训练集和测试集。

In the above code, we first load the 20 newsgroups dataset , which is a collection of newsgroup posts classified into different categories. We select four categories (alt.atheism, comp.graphics, sci.med, and soc.religion.christian) and split the data into training and testing sets.

然后，我们使用 scikit-learn 中的 CountVectorizer 类将文本数据转换成词袋表示。此表示计算文本中每个单词的出现次数，并将其表示为向量。

We then use the CountVectorizer class from scikit-learn to convert the text data into a bag-of-words representation. This representation counts the occurrence of each word in the text and represents it as a vector.

接下来，我们使用 fit() 方法训练多项式朴素贝叶斯分类器。此方法使用极大似然估计估计先验概率和给定类的情况下每个单词的可能性。然后，可以使用 predict() 方法使用测试集上的分类器进行预测。

Next, we train a Multinomial Naive Bayes classifier using the fit() method. This method estimates the prior probabilities and the likelihood of each word given the class using maximum likelihood estimation. The classifier can then be used to make predictions on the test set using the predict() method.

最后，我们使用 scikit-learn 中的 accuracy_score() 函数计算分类器的准确性。

Finally, we calculate the accuracy of the classifier using the accuracy_score() function from scikit-learn.

执行此代码时，将生成以下输出 −

When you execute this code, it will produce the following output −

Accuracy: 0.9340878828229028