贝叶斯学习(Bayesian Learning)提高篇

本文介绍了贝叶斯学习中的BayesOptimalClassifier和NaiveBayesClassifier,讨论了条件独立性和拉普拉斯平滑在处理数据中的作用。此外,还探讨了朴素贝叶斯在文档分类问题上的应用,并提到了贝叶斯网络的概念,强调了变量间的关系和联合概率分布的重要性。

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前言

本文将基于UoA的课件,连接上一篇博文介绍机器学习中的贝叶斯。看不太懂的读者请先阅读:贝叶斯学习(Bayesian Learning)基础篇

涉及的英语比较基础,所以为节省时间(不是full-time,还有其他三门课程,所以时间还是比较紧的),只在我以为需要解释的地方进行解释。

此文不用于任何商业用途,仅仅是个人学习过程笔记以及心得体会,侵必删。

Review Bayes Optimal Classifier

Bayes Optimal Classifier, also known as Bayes classifier, is a probabilistic model used for classification tasks. It is based on Bayes’ theorem and makes a decision based on the posterior probability of each class given the observed data.

The Bayes Optimal Classifier assumes that the distribution of the input data and the conditional probabilities of each class given the data are known. It then computes the posterior probability of each class given the data and chooses the class with the highest posterior probability as the classification result.

One of the advantages of the Bayes Optimal Classifier is that it is theoretically optimal, meaning it achieves the lowest possible classification error rate for any given distribution of the data. However, in practice, it may be difficult to estimate the distribution of the data accurately, and the assumptions made by the model may not hold.

The Bayes Optimal Classifier is often used as a benchmark for evaluating the performance of other classifiers, and it serves as a theoretical foundation for many probabilistic classification methods such as Naive Bayes, Bayesian Networks, and Bayesian Linear Regression.

Naive Bayes Classifier

在这里插入图片描述For the concept

The Naive Bayes Classifier is a probabilistic algorithm that is commonly used for classification tasks. It is based on the Bayes theorem and applies to learning tasks where each instance is described by a conjunction of attribute values and where the target function can take on any value from some finite set.

In other words, the Naive Bayes Classifier is used when we have a set of attributes that describe an instance, and we want to predict the value of a target variable based on these attributes. The attributes are assumed to be independent of each other, and this assumption is known as the “naive” assumption, hence the name “Naive Bayes”.

For example, consider a dataset of emails that are labeled as either spam or not spam. Each email is described by a set of attributes such as the presence of certain words or phrases, the length of the email, etc. The target variable is the label (spam or not spam) that we want to predict for new emails. The Naive Bayes Classifier can be used to learn a model from this data, which can then be used to predict the label of new emails based on their attributes.

在这里插入图片描述First, given the training data D, we can estimate the probability of each target value Vj by counting its frequency in D.

However, estimating the joint probability of all attribute values and target value P(a1, a2, …, an | Vj) is not feasible because the number of possible instances is too large (|all possible instances| x |V|).

To simplify the problem, the naive Bayes classifier assumes that the attribut

### 贝叶斯学习概述 贝叶斯学习是一种基于概率论的方法,在给定数据的情况下更新假设的概率。这种方法的核心在于利用先验知识和新观测的数据来调整对某一事件发生的信念程度。 #### 贝叶斯算法原理详解 贝叶斯理论提供了一种计算条件概率的方式,即在已知某些条件下另一些情况发生的可能性大小。具体来说,通过贝叶斯公式可以将先前的经验(称为先验分布)与当前观察到的新证据相结合,从而得出更精确的结果预测[^1]。 对于分类问题而言,当输入特征向量 \( \mathbf{x}=(x_1,x_2,\ldots,x_n)\),以及类别标签\( C_k\)时,目标是从所有可能的类中找到使得后验概率最大的那个: \[ P(C_i|\mathbf{x})=\frac{P(\mathbf{x}|C_i)P(C_i)}{\sum_jP(\mathbf{x}|C_j)P(C_j)}\] 这里的关键部分是似然函数\( P(\mathbf{x}|C_i)\),它描述了如果样本确实属于某个特定类别,则该样本出现的可能性;而分母则是为了使整个表达式的总和等于1所必需的比例因子。 #### 朴素贝叶斯分类器介绍 作为一种特殊的贝叶斯分类技术——朴素贝叶斯假定了各个属性之间相互独立这一简化前提下工作。这意味着即使两个或多个属性实际上存在关联性,在建模过程中也会被视作彼此无关处理。尽管这个假设通常并不完全成立,但在许多实际应用场景里仍然能够取得不错的效果[^3]。 下面给出一个简单的Python实现例子展示如何训练一个朴素贝叶斯分类器: ```python from sklearn.datasets import load_iris from sklearn.model_selection import train_test_split from sklearn.naive_bayes import GaussianNB # 加载鸢尾花数据集作为示例 data = load_iris() X_train, X_test, y_train, y_test = train_test_split(data.data, data.target, test_size=0.5, random_state=0) gnb = GaussianNB() # 创建高斯朴素贝叶斯实例 y_pred = gnb.fit(X_train, y_train).predict(X_test) print(f"Number of mislabeled points out of a total {X_test.shape[0]} points : {(y_test != y_pred).sum()}") ``` 此代码片段展示了如何使用Scikit-Learn库快速构建并评估一个基本的Gaussian Naive Bayes模型性能。 #### 应用案例分析 贝叶斯网络作为一个强大的工具,不仅限于简单分类任务的应用范围还包括但不限于以下几个方面: - **医疗诊断**:根据病人的症状和其他健康指标推断潜在疾病的存在与否; - **垃圾邮件过滤**:识别电子邮件是否为不受欢迎的信息流; - **自然语言处理**:解析文本内容意图或是情感倾向判断等任务上也有着广泛应用场景[^2]。
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