BAYESIAN INFERENCE FOR A SOFTWARE RELIABILITY MODEL USING METRICS INFORMATION.
Bayes(贝叶斯)估计

•
参数作为随机变量
• 条件分布: p(x1,x2,..xn | )
精选完整ppt课件
几个学派(3)
• 信念学派:
• 带头人:Fisher
• 观点:概率是频率
•
主观不是概率,而是信念度
•
参数不是随机变量,仅是普通变量
• 似然函数: L( | x1,x2,..xn)
精选完整ppt课件
批评1:置信区间
后验风险:
• Bayesian风险与后验风险
(L(,)p(x|) ()d)dx
• 后验分析最小=>Bayesian风险最小
精选完整ppt课件
两种常用损失函数:
• 平方损失:
L(,)()2
– 最小Bayesian风险估计:后验期望
• 点损失:
L(a,
)
0,|
a
|
1,|
a
|
– 最大后验密度估计
精选完整ppt课件
• 3、联合分布密度->条件分布密度
• p(x1,x2,..xn | ), 是随机变量
• 4、确定的先验分布() • 5、利用Bayesian公式求后验分布密度 • 6、使用后验分布做推断(参数估计、假设检验)
精选完整ppt课件
例1:两点分布b(1,p)的
• 1. 联合分布:p(x|)nxx(1)nx
• 使得 h ( |r ) p (x |)* ( )与先验分布同类型
• 若p(x|)服从正态分布,选正态分布 • 若p(x|)服从两点分布,选Beta分布 • 若p(x|)服从指数分布,选逆Gamma分布
精选完整ppt课件
Bayes统计推断问题
• 参数估计:
– 点估计 – 区间估计
人工智能领域中英文专有名词汇总

名词解释中英文对比<using_information_sources> social networks 社会网络abductive reasoning 溯因推理action recognition(行为识别)active learning(主动学习)adaptive systems 自适应系统adverse drugs reactions(药物不良反应)algorithm design and analysis(算法设计与分析) algorithm(算法)artificial intelligence 人工智能association rule(关联规则)attribute value taxonomy 属性分类规范automomous agent 自动代理automomous systems 自动系统background knowledge 背景知识bayes methods(贝叶斯方法)bayesian inference(贝叶斯推断)bayesian methods(bayes 方法)belief propagation(置信传播)better understanding 内涵理解big data 大数据big data(大数据)biological network(生物网络)biological sciences(生物科学)biomedical domain 生物医学领域biomedical research(生物医学研究)biomedical text(生物医学文本)boltzmann machine(玻尔兹曼机)bootstrapping method 拔靴法case based reasoning 实例推理causual models 因果模型citation matching (引文匹配)classification (分类)classification algorithms(分类算法)clistering algorithms 聚类算法cloud computing(云计算)cluster-based retrieval (聚类检索)clustering (聚类)clustering algorithms(聚类算法)clustering 聚类cognitive science 认知科学collaborative filtering (协同过滤)collaborative filtering(协同过滤)collabrative ontology development 联合本体开发collabrative ontology engineering 联合本体工程commonsense knowledge 常识communication networks(通讯网络)community detection(社区发现)complex data(复杂数据)complex dynamical networks(复杂动态网络)complex network(复杂网络)complex network(复杂网络)computational biology 计算生物学computational biology(计算生物学)computational complexity(计算复杂性) computational intelligence 智能计算computational modeling(计算模型)computer animation(计算机动画)computer networks(计算机网络)computer science 计算机科学concept clustering 概念聚类concept formation 概念形成concept learning 概念学习concept map 概念图concept model 概念模型concept modelling 概念模型conceptual model 概念模型conditional random field(条件随机场模型) conjunctive quries 合取查询constrained least squares (约束最小二乘) convex programming(凸规划)convolutional neural networks(卷积神经网络) customer relationship management(客户关系管理) data analysis(数据分析)data analysis(数据分析)data center(数据中心)data clustering (数据聚类)data compression(数据压缩)data envelopment analysis (数据包络分析)data fusion 数据融合data generation(数据生成)data handling(数据处理)data hierarchy (数据层次)data integration(数据整合)data integrity 数据完整性data intensive computing(数据密集型计算)data management 数据管理data management(数据管理)data management(数据管理)data miningdata mining 数据挖掘data model 数据模型data models(数据模型)data partitioning 数据划分data point(数据点)data privacy(数据隐私)data security(数据安全)data stream(数据流)data streams(数据流)data structure( 数据结构)data structure(数据结构)data visualisation(数据可视化)data visualization 数据可视化data visualization(数据可视化)data warehouse(数据仓库)data warehouses(数据仓库)data warehousing(数据仓库)database management systems(数据库管理系统)database management(数据库管理)date interlinking 日期互联date linking 日期链接Decision analysis(决策分析)decision maker 决策者decision making (决策)decision models 决策模型decision models 决策模型decision rule 决策规则decision support system 决策支持系统decision support systems (决策支持系统) decision tree(决策树)decission tree 决策树deep belief network(深度信念网络)deep learning(深度学习)defult reasoning 默认推理density estimation(密度估计)design methodology 设计方法论dimension reduction(降维) dimensionality reduction(降维)directed graph(有向图)disaster management 灾害管理disastrous event(灾难性事件)discovery(知识发现)dissimilarity (相异性)distributed databases 分布式数据库distributed databases(分布式数据库) distributed query 分布式查询document clustering (文档聚类)domain experts 领域专家domain knowledge 领域知识domain specific language 领域专用语言dynamic databases(动态数据库)dynamic logic 动态逻辑dynamic network(动态网络)dynamic system(动态系统)earth mover's distance(EMD 距离) education 教育efficient algorithm(有效算法)electric commerce 电子商务electronic health records(电子健康档案) entity disambiguation 实体消歧entity recognition 实体识别entity recognition(实体识别)entity resolution 实体解析event detection 事件检测event detection(事件检测)event extraction 事件抽取event identificaton 事件识别exhaustive indexing 完整索引expert system 专家系统expert systems(专家系统)explanation based learning 解释学习factor graph(因子图)feature extraction 特征提取feature extraction(特征提取)feature extraction(特征提取)feature selection (特征选择)feature selection 特征选择feature selection(特征选择)feature space 特征空间first order logic 一阶逻辑formal logic 形式逻辑formal meaning prepresentation 形式意义表示formal semantics 形式语义formal specification 形式描述frame based system 框为本的系统frequent itemsets(频繁项目集)frequent pattern(频繁模式)fuzzy clustering (模糊聚类)fuzzy clustering (模糊聚类)fuzzy clustering (模糊聚类)fuzzy data mining(模糊数据挖掘)fuzzy logic 模糊逻辑fuzzy set theory(模糊集合论)fuzzy set(模糊集)fuzzy sets 模糊集合fuzzy systems 模糊系统gaussian processes(高斯过程)gene expression data 基因表达数据gene expression(基因表达)generative model(生成模型)generative model(生成模型)genetic algorithm 遗传算法genome wide association study(全基因组关联分析) graph classification(图分类)graph classification(图分类)graph clustering(图聚类)graph data(图数据)graph data(图形数据)graph database 图数据库graph database(图数据库)graph mining(图挖掘)graph mining(图挖掘)graph partitioning 图划分graph query 图查询graph structure(图结构)graph theory(图论)graph theory(图论)graph theory(图论)graph theroy 图论graph visualization(图形可视化)graphical user interface 图形用户界面graphical user interfaces(图形用户界面)health care 卫生保健health care(卫生保健)heterogeneous data source 异构数据源heterogeneous data(异构数据)heterogeneous database 异构数据库heterogeneous information network(异构信息网络) heterogeneous network(异构网络)heterogenous ontology 异构本体heuristic rule 启发式规则hidden markov model(隐马尔可夫模型)hidden markov model(隐马尔可夫模型)hidden markov models(隐马尔可夫模型) hierarchical clustering (层次聚类) homogeneous network(同构网络)human centered computing 人机交互技术human computer interaction 人机交互human interaction 人机交互human robot interaction 人机交互image classification(图像分类)image clustering (图像聚类)image mining( 图像挖掘)image reconstruction(图像重建)image retrieval (图像检索)image segmentation(图像分割)inconsistent ontology 本体不一致incremental learning(增量学习)inductive learning (归纳学习)inference mechanisms 推理机制inference mechanisms(推理机制)inference rule 推理规则information cascades(信息追随)information diffusion(信息扩散)information extraction 信息提取information filtering(信息过滤)information filtering(信息过滤)information integration(信息集成)information network analysis(信息网络分析) information network mining(信息网络挖掘) information network(信息网络)information processing 信息处理information processing 信息处理information resource management (信息资源管理) information retrieval models(信息检索模型) information retrieval 信息检索information retrieval(信息检索)information retrieval(信息检索)information science 情报科学information sources 信息源information system( 信息系统)information system(信息系统)information technology(信息技术)information visualization(信息可视化)instance matching 实例匹配intelligent assistant 智能辅助intelligent systems 智能系统interaction network(交互网络)interactive visualization(交互式可视化)kernel function(核函数)kernel operator (核算子)keyword search(关键字检索)knowledege reuse 知识再利用knowledgeknowledgeknowledge acquisitionknowledge base 知识库knowledge based system 知识系统knowledge building 知识建构knowledge capture 知识获取knowledge construction 知识建构knowledge discovery(知识发现)knowledge extraction 知识提取knowledge fusion 知识融合knowledge integrationknowledge management systems 知识管理系统knowledge management 知识管理knowledge management(知识管理)knowledge model 知识模型knowledge reasoningknowledge representationknowledge representation(知识表达) knowledge sharing 知识共享knowledge storageknowledge technology 知识技术knowledge verification 知识验证language model(语言模型)language modeling approach(语言模型方法) large graph(大图)large graph(大图)learning(无监督学习)life science 生命科学linear programming(线性规划)link analysis (链接分析)link prediction(链接预测)link prediction(链接预测)link prediction(链接预测)linked data(关联数据)location based service(基于位置的服务) loclation based services(基于位置的服务) logic programming 逻辑编程logical implication 逻辑蕴涵logistic regression(logistic 回归)machine learning 机器学习machine translation(机器翻译)management system(管理系统)management( 知识管理)manifold learning(流形学习)markov chains 马尔可夫链markov processes(马尔可夫过程)matching function 匹配函数matrix decomposition(矩阵分解)matrix decomposition(矩阵分解)maximum likelihood estimation(最大似然估计)medical research(医学研究)mixture of gaussians(混合高斯模型)mobile computing(移动计算)multi agnet systems 多智能体系统multiagent systems 多智能体系统multimedia 多媒体natural language processing 自然语言处理natural language processing(自然语言处理) nearest neighbor (近邻)network analysis( 网络分析)network analysis(网络分析)network analysis(网络分析)network formation(组网)network structure(网络结构)network theory(网络理论)network topology(网络拓扑)network visualization(网络可视化)neural network(神经网络)neural networks (神经网络)neural networks(神经网络)nonlinear dynamics(非线性动力学)nonmonotonic reasoning 非单调推理nonnegative matrix factorization (非负矩阵分解) nonnegative matrix factorization(非负矩阵分解) object detection(目标检测)object oriented 面向对象object recognition(目标识别)object recognition(目标识别)online community(网络社区)online social network(在线社交网络)online social networks(在线社交网络)ontology alignment 本体映射ontology development 本体开发ontology engineering 本体工程ontology evolution 本体演化ontology extraction 本体抽取ontology interoperablity 互用性本体ontology language 本体语言ontology mapping 本体映射ontology matching 本体匹配ontology versioning 本体版本ontology 本体论open government data 政府公开数据opinion analysis(舆情分析)opinion mining(意见挖掘)opinion mining(意见挖掘)outlier detection(孤立点检测)parallel processing(并行处理)patient care(病人医疗护理)pattern classification(模式分类)pattern matching(模式匹配)pattern mining(模式挖掘)pattern recognition 模式识别pattern recognition(模式识别)pattern recognition(模式识别)personal data(个人数据)prediction algorithms(预测算法)predictive model 预测模型predictive models(预测模型)privacy preservation(隐私保护)probabilistic logic(概率逻辑)probabilistic logic(概率逻辑)probabilistic model(概率模型)probabilistic model(概率模型)probability distribution(概率分布)probability distribution(概率分布)project management(项目管理)pruning technique(修剪技术)quality management 质量管理query expansion(查询扩展)query language 查询语言query language(查询语言)query processing(查询处理)query rewrite 查询重写question answering system 问答系统random forest(随机森林)random graph(随机图)random processes(随机过程)random walk(随机游走)range query(范围查询)RDF database 资源描述框架数据库RDF query 资源描述框架查询RDF repository 资源描述框架存储库RDF storge 资源描述框架存储real time(实时)recommender system(推荐系统)recommender system(推荐系统)recommender systems 推荐系统recommender systems(推荐系统)record linkage 记录链接recurrent neural network(递归神经网络) regression(回归)reinforcement learning 强化学习reinforcement learning(强化学习)relation extraction 关系抽取relational database 关系数据库relational learning 关系学习relevance feedback (相关反馈)resource description framework 资源描述框架restricted boltzmann machines(受限玻尔兹曼机) retrieval models(检索模型)rough set theroy 粗糙集理论rough set 粗糙集rule based system 基于规则系统rule based 基于规则rule induction (规则归纳)rule learning (规则学习)rule learning 规则学习schema mapping 模式映射schema matching 模式匹配scientific domain 科学域search problems(搜索问题)semantic (web) technology 语义技术semantic analysis 语义分析semantic annotation 语义标注semantic computing 语义计算semantic integration 语义集成semantic interpretation 语义解释semantic model 语义模型semantic network 语义网络semantic relatedness 语义相关性semantic relation learning 语义关系学习semantic search 语义检索semantic similarity 语义相似度semantic similarity(语义相似度)semantic web rule language 语义网规则语言semantic web 语义网semantic web(语义网)semantic workflow 语义工作流semi supervised learning(半监督学习)sensor data(传感器数据)sensor networks(传感器网络)sentiment analysis(情感分析)sentiment analysis(情感分析)sequential pattern(序列模式)service oriented architecture 面向服务的体系结构shortest path(最短路径)similar kernel function(相似核函数)similarity measure(相似性度量)similarity relationship (相似关系)similarity search(相似搜索)similarity(相似性)situation aware 情境感知social behavior(社交行为)social influence(社会影响)social interaction(社交互动)social interaction(社交互动)social learning(社会学习)social life networks(社交生活网络)social machine 社交机器social media(社交媒体)social media(社交媒体)social media(社交媒体)social network analysis 社会网络分析social network analysis(社交网络分析)social network(社交网络)social network(社交网络)social science(社会科学)social tagging system(社交标签系统)social tagging(社交标签)social web(社交网页)sparse coding(稀疏编码)sparse matrices(稀疏矩阵)sparse representation(稀疏表示)spatial database(空间数据库)spatial reasoning 空间推理statistical analysis(统计分析)statistical model 统计模型string matching(串匹配)structural risk minimization (结构风险最小化) structured data 结构化数据subgraph matching 子图匹配subspace clustering(子空间聚类)supervised learning( 有support vector machine 支持向量机support vector machines(支持向量机)system dynamics(系统动力学)tag recommendation(标签推荐)taxonmy induction 感应规范temporal logic 时态逻辑temporal reasoning 时序推理text analysis(文本分析)text anaylsis 文本分析text classification (文本分类)text data(文本数据)text mining technique(文本挖掘技术)text mining 文本挖掘text mining(文本挖掘)text summarization(文本摘要)thesaurus alignment 同义对齐time frequency analysis(时频分析)time series analysis( 时time series data(时间序列数据)time series data(时间序列数据)time series(时间序列)topic model(主题模型)topic modeling(主题模型)transfer learning 迁移学习triple store 三元组存储uncertainty reasoning 不精确推理undirected graph(无向图)unified modeling language 统一建模语言unsupervisedupper bound(上界)user behavior(用户行为)user generated content(用户生成内容)utility mining(效用挖掘)visual analytics(可视化分析)visual content(视觉内容)visual representation(视觉表征)visualisation(可视化)visualization technique(可视化技术) visualization tool(可视化工具)web 2.0(网络2.0)web forum(web 论坛)web mining(网络挖掘)web of data 数据网web ontology lanuage 网络本体语言web pages(web 页面)web resource 网络资源web science 万维科学web search (网络检索)web usage mining(web 使用挖掘)wireless networks 无线网络world knowledge 世界知识world wide web 万维网world wide web(万维网)xml database 可扩展标志语言数据库附录 2 Data Mining 知识图谱(共包含二级节点15 个,三级节点93 个)间序列分析)监督学习)领域 二级分类 三级分类。
贝叶斯算法(bayesian)介绍

贝叶斯过滤算法举例
综合考虑两个哈希表,共有四个 TOKEN 串:法 、轮、 功、 律 当邮件中出现“法”时,该邮件为垃圾邮 件的概率为: P = 0.3/ ( 0.3 + 0.5 ) = 0.375
出现“轮”时,该邮件为垃圾邮件的概率 为: P = 0.3/ ( 0.3 + 0 ) = 1
贝叶斯过滤算法的主要步骤
算法的基本思想: 根据已有的垃圾邮件和非垃圾邮件建立 贝叶斯概率库,利用贝叶斯概率库分析 预测新邮件为垃圾邮件的概率 优点: 1、纯粹根据统计学规律运作 2、可计算性强
贝叶斯过滤算法的主要步骤
1. 收集大量的垃圾邮件和非垃圾邮件, 建立垃圾邮件集和非垃圾邮件集。
t1,t2 …….tn 代表 TOห้องสมุดไป่ตู้EN 串
则 P ( A|ti )表示在邮件中出现 TOKEN 串 ti 时,该邮件 为垃圾邮件的概率。 设 P1 ( ti ) = ( ti 在 hashtable_good 中的值) P2 ( ti ) = ( ti 在 hashtable_ bad 中的值) 则 P ( A|ti ) =P2 ( ti ) /[ ( P1 ( ti ) +P2 ( ti ) ] ;
贝叶斯过滤算法举例
例: 你的邮箱里现有两份邮件: 一封是含有 “ 法轮功 ” 字样的垃圾邮件 A 另一封是含有 “ 法律 ” 字样的非垃圾邮 件B。
贝叶斯过滤算法举例
下面根据这两份已有邮件来建立贝叶斯 概率库 根据邮件 A 生成 hashtable_bad ,该哈 希表中的记录为 法: 1 次 轮: 1 次 功: 1 次
2. 提取邮件主题和邮件体中的独立字符 串,例如 ABC32,¥234等作为TOKEN 串并统计提取出的TOKEN串出现的次 数即字频。按照上述的方法分别处理垃 圾邮件集和非垃圾邮件集中的所有邮件。
数据挖掘运用bayes算法进行分类预测题目

数据挖掘运用bayes算法进行分类预测题目《数据挖掘:运用Bayes算法进行分类预测》1. 引言数据挖掘作为一种重要的数据分析工具,应用广泛,其中分类预测是其中的一个重要分支。
Bayes算法作为经典的分类预测方法之一,在实际应用中具有重要的意义。
本文将针对数据挖掘中Bayes算法的分类预测进行深入探讨,帮助读者更深入地理解该主题。
2. 数据挖掘的基本概念数据挖掘是一种从大量数据中发现未知的模式和规律的过程,旨在为决策提供有用的信息支持。
这些模式和规律可以帮助用户发现数据的内在联系,辅助用户进行决策和未来发展的规划。
分类预测是数据挖掘中的一个重要任务,它通过对已知类别的样本进行学习,然后根据学习到的模型对新样本进行分类预测。
3. Bayes算法的原理和应用Bayes算法是一种基于概率的分类方法,它基于贝叶斯定理,利用已知的样本信息来预测新的样本所属的类别。
具体而言,Bayes算法假设每个特征与类别之间相互独立,并基于该假设计算每个类别下样本的概率,并根据概率大小确定样本所属的类别。
该算法简单易懂,且在很多实际应用中取得了良好的效果,因此备受青睐。
4. Bayes算法在分类预测中的应用Bayes算法在各个领域都有广泛的应用,例如垃圾邮件识别、疾病预测、情感分析等。
在垃圾邮件识别中,Bayes算法可以通过分析邮件内容和发件人等信息,判断一封邮件是否是垃圾邮件;在疾病预测中,可以根据患者的各项指标和病史,预测患者是否患有某种疾病;在情感分析中,可以根据文本内容判断其所表达的情感是积极的、消极的还是中立的。
这些应用充分展示了Bayes算法在分类预测中的重要作用。
5. 个人观点和理解个人认为Bayes算法作为一种基于概率的分类方法,其在数据挖掘中具有重要的应用价值。
它简单易懂,计算效率高,且在很多实际应用中表现出色。
然而,Bayes算法也有其局限性,例如对特征之间的独立性要求较高,且在样本类别不平衡时表现较差。
贝叶斯分类英文缩写

贝叶斯分类英文缩写Bayesian classification, often abbreviated as "Naive Bayes," is a popular machine learning algorithm used for classification tasks. It is based on Bayes' theorem and assumes that features are independent of each other, hence the "naive" aspect. 贝叶斯分类,通常缩写为“朴素贝叶斯”,是一种常用的用于分类任务的机器学习算法。
它基于贝叶斯定理,并假设特征相互独立,因此有“朴素”之称。
One of the main advantages of Naive Bayes classification is its simplicity and efficiency. It is easy to implement and works well with large datasets. Additionally, it performs well even with few training examples. However, its main downside is the assumption of feature independence, which may not hold true in real-world scenarios. 朴素贝叶斯分类的主要优点之一是其简单和高效。
它易于实现,适用于大型数据集。
此外,即使只有少量训练样本,它也能表现良好。
然而,其主要缺点是特征独立的假设,在真实场景中可能并不成立。
From a mathematical perspective, Naive Bayes classification calculates the probability of each class given a set of features using Bayes' theorem. It estimates the likelihood of each class based on thetraining data and the probabilities of different features belonging to each class. The class with the highest probability is assigned to the input data point. 从数学角度来看,朴素贝叶斯分类使用贝叶斯定理计算了给定一组特征时每个类别的概率。
- 贝叶斯近似算法介绍

- 贝叶斯近似算法介绍全文共四篇示例,供读者参考第一篇示例:贝叶斯近似算法(Bayesian Approximation Algorithm)是一种基于贝叶斯统计推断原理的近似算法,通常用于解决模型复杂、数据量大的问题。
在机器学习领域中,贝叶斯方法是一种常见且有效的方法,它不仅可以用于分类、回归等监督学习任务,还可以应用于聚类、降维、推荐系统等无监督学习任务。
贝叶斯近似算法的核心思想是基于贝叶斯定理进行概率推断,通过对参数的后验分布进行近似推断,从而得到参数的估计结果。
与传统的最大似然估计方法相比,贝叶斯方法能够更好地利用先验知识,对参数的不确定性进行更合理的建模,同时还能够避免过拟合的问题。
在实际应用中,由于后验分布的计算通常是非常困难甚至不可行的,因此需要借助于贝叶斯近似算法来进行推断。
常见的贝叶斯近似算法包括马尔科夫链蒙特卡洛(MCMC)方法、变分推断方法、拉普拉斯近似方法等。
马尔可夫链蒙特卡洛方法是一种基于随机模拟的推断方法,通过构建马尔可夫链来模拟参数的后验分布。
通过多次迭代采样,最终得到参数的后验分布的近似值。
但是MCMC方法的计算复杂度较高,收敛速度较慢,在处理大规模数据时可能会面临挑战。
变分推断方法是另一种常见的贝叶斯近似算法,它通过最优化一个参数化的分布来近似真实的后验分布。
变分推断方法通常会引入一些近似假设,例如独立性假设、指数族假设等,从而简化推断的计算复杂度。
变分推断方法的优点是计算效率高,但是可能会引入一定的偏差。
拉普拉斯近似方法是一种基于高斯分布的近似推断方法,通过在后验分布的峰值处进行局部近似,得到参数的估计结果。
拉普拉斯近似方法通常适用于后验分布近似是单峰分布的情况,当后验分布是多峰分布时可能会出现不准确的情况。
贝叶斯近似算法是一种在处理复杂、大规模数据时非常有效的推断方法。
通过合理地选择适当的近似算法,结合先验知识和数据信息,可以得到更加准确和稳健的模型参数估计结果。
python 贝叶斯拟合-概述说明以及解释

python 贝叶斯拟合-概述说明以及解释1.引言1.1 概述在数据分析和机器学习领域中,拟合是一项重要的任务,用于找到一条曲线或函数来拟合已知的数据点。
贝叶斯拟合是一种基于贝叶斯统计理论的拟合方法,通过使用先验概率和后验概率来估计模型参数的不确定性。
贝叶斯拟合与传统的最小二乘拟合方法有所不同。
最小二乘拟合仅仅关注于找到与数据点之间的最小误差的函数,而贝叶斯拟合方法通过引入先验概率来更全面地分析参数的分布情况。
在传统的拟合方法中,参数的选择通常是基于一些假设或者经验规则。
而贝叶斯拟合方法不需要依赖于这些假设,它可以根据数据本身来调整模型参数,并且在推理过程中提供了对参数估计的不确定性的度量。
Python中提供了多种贝叶斯拟合的方法和工具,例如使用PyMC3库进行贝叶斯线性回归拟合。
通过使用这些工具,我们可以轻松地进行贝叶斯拟合分析,探索参数的分布情况并对拟合结果进行评估。
本文将介绍贝叶斯拟合的基本概念,并详细介绍Python中的贝叶斯拟合方法。
我们将以贝叶斯线性回归为例,演示如何使用PyMC3库进行贝叶斯拟合,并展示拟合结果的可视化和参数不确定性的分析。
在结论部分,我们将对贝叶斯拟合的方法进行总结,并讨论其在数据分析和机器学习领域中的应用前景。
贝叶斯拟合方法在处理不确定性、决策分析和模型比较等方面具有重要的优势,有望在未来得到更广泛的应用和研究。
文章结构部分的内容应该描述整篇文章的结构和组织方式,以帮助读者更好地理解和阅读文章。
下面是关于文章结构部分的内容示例:1.2 文章结构本文主要介绍了Python中的贝叶斯拟合方法及其应用。
为了帮助读者更好地理解,本文按照以下结构进行组织和阐述。
首先,我们将在引言部分对本文进行概述,以便读者对贝叶斯拟合有一个整体的了解。
接着,我们将详细介绍贝叶斯拟合的基本概念,包括其原理、应用场景等内容。
在这一部分中,我们将通过理论知识的讲解,使读者对贝叶斯拟合有一个清晰的认识。
Bayesian Inference

p(black) = 18/38 = 0.473
p(red) = 18/38 = 0.473
p(color )
What is a conditional probability?
p(black|col1) = 6/12 = 0.5
p(color | col1)
p(black|col2) = 8/12 = 0.666
p(x)
Bayesian Roulette
• We’re interested in which column will win. • p(column) is our prior. • We learn color=black.
Bayesian Roulette
• • • • We’re interested in which column will win. p(column) is our prior. We learn color=black. What is p(color=black|column)?
What do you need to know to use it?
• You need to be able to express your prior beliefs about x as a probability distribution, p (x ) • You must able to relate your new evidence to your variable of interest in terms of it’s likelihood, p(y|x) • You must be able to multiply.
p(black|col1) = 6/12 = 0.5 p(black|col2) = 8/12 = 0.666 p(black|col3) = 4/12 = 0.333 p(black|zeros) = 0/2 = 0
- 1、下载文档前请自行甄别文档内容的完整性,平台不提供额外的编辑、内容补充、找答案等附加服务。
- 2、"仅部分预览"的文档,不可在线预览部分如存在完整性等问题,可反馈申请退款(可完整预览的文档不适用该条件!)。
- 3、如文档侵犯您的权益,请联系客服反馈,我们会尽快为您处理(人工客服工作时间:9:00-18:30)。
Working Paper 01-20Statistics and Econometrics Series 14 March 2001Departamento de Estadística y Econometría Universidad Carlos III de MadridCalle Madrid, 12628903 Getafe (Spain)Fax (34) 91 624-98-49BAYESIAN INFERENCE FOR A SOFTWARE RELIABILITY MODEL USINGMETRICS INFORMATION.M.P. Wiper and M.T. Rodríguez Bernal*AbstractIn this paper, we are concerned with predicting the number of faults N and the time to next failure of a piece of software. Information in the form of software metrics data is used to estimate the prior distribution of N via a Poisson regression model. Given failure time data, and a well known model for software failures, we show how to sample the posterior distribution using Gibbs sampling, as implemented in the package "WinBugs". The approach is illustrated with a practical example.Keywords: Software metrics; principal components; software reliability; Jelinski Moranda model; Bayesian inference; Gibbs sampling; WinBugs.*Wiper, Departamento de Estadística y Econometría, Universidad Carlos III de Madrid, C/ Madrid 126, 28903 Getafe -Madrid-, Tfno: 91-6249852, e-mail: mwiper@est-econ.uc3m.es; Rodríguez, Departamento de Estadística y Econometría, Universidad Carlos III de Madrid, e-mail: mrodrigu@est-econ.uc3m.es. The work of Mike Wiper has been partially supported by the project Análisis Estadístico de Grandes Bancos de Datos Económicos y Empresariales con Estructura Compleja funded by the Spanish Ministry of Science and Technology.Bayesian inference for a software reliability model using metrics informationM P Wiper M T Rodriguez Bernal¤March8,2001AbstractIn this paper,we are concerned with predicting the number of faults N and the time to next failure of a piece of r-mation in the form of software metrics data is used to estimate theprior distribution of N via a Poisson regression model.Given failuretime data,and a well known model for software failures,we show howto sample the posterior distribution using Gibbs sampling,as imple-mented in the package“WinBugs”.The approach is illustrated witha practical example.Keywords:software metrics,principal components,software reli-ability,Jelinski Moranda model,Bayesian inference,Gibbs sampling,WinBugs.1IntroductionSoftware reliability can be de…ned as“the probability of failure-free oper-ation of a computer code for a speci…ed mission time in a speci…ed input environment”,(Singpurwalla and Wilson1999).The majority of software reliability models are concerned with the predic-tion of interfailure times,say T1;T2;:::,see for example Littlewood(1989) or Singpurwalla and Wilson(1994).It would seem plausible that failure times are related to the number of faults(or bugs)in the software,say N, and various models have been developed which are based on a bug counting approach.Speci…c examples are the models of Jelinski and Moranda(1972),¤Departamento de Estadística y Econometría,Universidad Carlos III de Madrid.1Goel and Okumoto(1978),Schick and Wolverton(1978)and Musa and Oku-moto(1987).See Singpurwalla and Wilson(1999)for a general review and comments about alternative approaches to failure time prediction.A second method for reliability assessment,which we do not consider in this article,uses random or partition testing methods to estimate the failure rate or probability that an input to the program from the operational pro…le will produce an erroneous output.See for example Musa(1993)and Hierons and Wiper(1997).A third approach to assessing software reliability,particularly in the soft-ware development process,concentrates on the measurement of character-istics of a piece of software,called software metrics.The most well known software metrics are probably the simple lines of code(LOC)and McCabe’s (1976)cyclomatic complexity measure1.A great many other metrics have also been de…ned.See,for example,Fenton and P‡eeger(1997)for a review.Various models have been used in order to predict the number of bugs (or other measures of quality such as cost,amendements etc.)in a piece of software using software metrics data.For example,Mayer and Sykes(1991), Wiper et al(1992)ed(classical)logistic regression approaches,normal regression models have also been used(Compton and Withrow1990)and Khoshgoftaar et al(1995)introduced a neural net approach.In this paper we are concerned with prediction of the number of bugs in a piece of software and the time to next failure given both failure time infor-mation and metrics data.Our objective is to introduce Bayesian inference techniques in order to combine the two sources of information.In Section2,we introduce on of the most well known software reliability models and comment on Bayesian inference for this model.In Section3, we introduce a Bayesian logistic regression model for the number number of faults in a piece of software and combine this model with that of the previous section and illustrate how to carry out Bayesian inference using Gibbs sampling.In Section4,we illustrate our approach with an example using real software metrics data and in Section5we consider some possible extensions.2A simple model for software faluresThe Jelinski Moranda(1972)model for software reliability is one of the sim-plest.It supposes that the distribution of the j’th interfailure time is given 1This measure is based on the representation of the code as a directed graph and is equal to be the number of edges minus the number of nodes plus2.2byT j j N;Á»E((N¡j+1)Á);an exponential distribution with mean1=(N¡j+1)Á.The underlying(possibly unrealistic)assumptions of this model are:1.The program contains an initial number of faults N.2.Each bug contributes the same amount to the failure rate.3.After each observed failure,a bug is detected and corrected.Bayesian inference for the Jelinski Moranda model is considered by,for example,Meinhold and Singpurwalla(1983)who introduce the following nat-ural independent prior distributions for N andÁ:N»P(¸)a Poisson distributionÁ»G(®;º)a gamma densityGiven the observation of n failures,then the likelihood function is,for N¸n,l(N;Áj data)/N!(N¡n)!Án exp0@¡n X j=1(N¡j+1)t jÁ1Aso that,for example,the full conditional posterior distributions are easily evaluated:N¡n j data;Á»P(¸exp(¡n¹tÁ))where¹t=1P n j=1t jÁj data;N»G0@®+n;º+n X j=1(N¡j+1)t j1A:Thus,it is easy to set up a Gibbs sampling scheme to sample the posterior distribution of(N;Á)2One problem with this model is the possible sensitivity to the election of the prior distribution,see,e.g Wiper et al(1998),Wilson and Wiper (2000).In particular,it is unclear how to elect the prior parameters¸;®;º. Given little prior knowledge,it would be natural to elect an uninformative, improper,prior distribution for(N;Á)but it is easy to show that in this case,the posterior distribution is also improper,see e.g.Wilson and Wiper (2000).In the following section,we illustrate how software metrics data might be used to provide information about the prior distribution for N.2The marginal posteriors of N andÁcan also be derived up to truncation of a one dimensional summation.33Software MetricsIn this section,we consider…rst a di¤erent problem,that of predicting the number of faults in a software program given metrics data.Suppose that,for a collection of programs m,we calculate k software metrics X i=(X i1;:::;X ik),where i=1;:::;m.Let N i be the number of faults in program i.Then the following logistic regression is a natural model relating faults to software metrics:N i j¸i»P(¸i)as in the previous sectionlog¸i=¯0+X i1¯1+:::+X ik¯kGiven fault data N1=n1;:::;N m=n m;various authors have used classi-cal logistic regression to estimate the unknown parameters¯,see e.g.Mayer and Sykes(1991),Wiper et al(1992).In this case we consider a Bayesian approach,which,as far as we know,has not been considered previously for metrics data.Given a prior distribution for¯,for example a normal distribution¯»N(b;B)then it can be shown that all of the conditional posterior parameter distributions are log concave(see Dellaportas and Smith1993)and so the adaptive rejection algorithm(Gilks and Wild1992)can be used to sample from the posterior distribution.This scheme is preprogrammed into the Gibbs sampling package WinBugs(Spiegelhalter et al1999).One problem with metrics data is that typically,software metrics are very highly correlated with program size,see e.g.Wiper et al(1992).Thus, rather than use the raw metrics data,to avoid problems of colinearity,it is normal to transform the metrics using,for example,principal components. Typically,we will have little prior information concerning these components and thus it is normal to use a relatively di¤use prior distribution for¯.Assuming that both sources of information are available,the logistic re-gression model for metrics data can be combined with the Jelinski Moranda model for failure times.Figure1shows a directed graph(a”Doodle”in”Winbugs”)which illustrates the dependence structure of the combined model.(In the diagram,we have used N new instead of N for the number of bugs we are trying to predict.)Using the graph,distributional assumptions for each node can be speci…ed within Winbugs.We illustrate the approach with an example using real metrics data and simulated failure time data in the following section.44ExampleTen metrics(Basili Hutchens,depth of nesting,LOC,McCabe,YAM etc.) and the number of amendments were collected from36unstructured software programs.A simple correlations analysis showed that the metrics were very highly correlated and thus,a principal components analysis was undertaken, which showed that virtually all(99%)of the variance in the metrics data was explained by the…rst…ve principal components.Here we use amendments to represent faults and assume the logistic re-gression model of section3for the mean number of amendments where the vector X contains the…rst…ve principal components and¯=(¯0;:::;¯5). Independent,relatively uninformative,normal distributions with mean0and precision10¡5were chosen for each¯i.5interfailure times were also generated using the Jelinski Moranda model (withÁ=0:1)from a program containing9amendments/bugs.A relatively uninformative gamma prior distribution G(:001;:001)was5used forÁ:A Gibbs run of5000iterations to”burn in”the chain and25000 iterations in equilibrium was used.(Using the usual checks,the chain could be seen to have coverged before5000iterations).In this case,the posterior mean estimate for¯coincides almost exactly with the classical logistic regression estimate based on the36metric data, which we would expect given the use of noninformative prior distributions. In Figures2and3,we illustrate the estimated posterior densities for N (N new)andÁ:Values of N between6and9have posterior probabilities greater than 0.1.The posterior mean is E[N j data]'9:4(standard deviation3.7)and a 95%probability interval for N is[5,19].Thus,the true value of N has been well estimated given the metrics and failure data.Figure3:Posterior density ofÁ:6The posterior mean of Áis E [Áj data ]'0:102(standard deviation 0:072)which again coincides very well with the true generating value.Finally,Figure 4illustrates the reliability function for the time to next (sixth)failure of the program:P (T 6>t j datos ):It can be seen that there is a positive probability that the program will not fail again,which is equal to the posterior probability that the program contained only the …ve discovered faults.t P (T >t )0.00.20.40.60.81.0Figure 4:Reliability function for the time to next failure.5Conclusions and ExtensionsIn this paper we have shown how Bayesian inference concerning the number of bugs and the time to next failure of a piece of software can be carried out given both failure time and software metrics data.Here we have supposed the Jelinski Moranda model for failure times,but the same basic techniques may be applied to other models.Work is currently in progress on Bayesian inference for these models.When there are a selection of possible models,the problem of model choice becomes important.Theoretically,di¤erent models may be assessed from a Bayesian perspective via the use of Bayes factors,see e.g.Kass and Raftery (1995).A practical problem is that Bayes factors are di¢cult to calculate solely from the WinBugs output and thus,the programming task is somewhat more complicated.76AcknowledgementsThe work of Mike Wiper has been partially supported by the project Análi-sis Estadístico de Grandes Bancos de Datos Económicos y Empresariales con Estructura Compleja funded by the Spanish Ministry of Science and Tech-nology.BibliographyCompton,J.and Withrow,C.(1990).Prediction and control of Ada soft-ware defects.Journal of Systems and Software,12,199-207.Dellaportas,P.and Smith,A.F.M.(1993).Bayesian inference for general-ized linear and proportional hazards models via Gibbs sampling.Ap-plied Statistics,42,443-459.Fenton,N.E.and P‡eeger,S.L.(1997).Software Metrics.A Rigorous and Practical Approach,2nd.edition.Boston:PWS Publishing.Gilks,W.R.and Wild,P.(1992).Adaptive rejection sampling for Gibbs sampling.Applied Statistics,41,337-348.Goel,A.L.and Okumoto,K.(1979).Time dependent error detection rate model for software reliability and other performance measures.IEEE Trans.Rel.,R-28,206-211.Hierons,R.M.and Wiper,M.P.(1997).Estimation of Failure Rate using Random and Partition Testing.Journal of Software Testing,Veri…ca-tion and Reliability,7,153-164.Jelinski,Z.and Moranda,P.(1972).Software reliability research.In Sta-tistical Computer Performance Evaluation,W.Freiberger,editor,465–484.New York:Academic Press.IEEE Trans.Reliability,R-34,216–218.Kass,R.E.and Raftery,A.E.(1995).Bayes Factors.J.Amer.Statist.Assoc.,90,773-795.Khoshgoftaar,T.M.,Panday,A.S.and Lanning,D.L.(1995).Application of nueral networks for predicting program faults.Annals of Software Engineering,1,141-154.Littlewood,B.(1989).Forecasting Software Reliability.Lecture Notes in Computer Science,No.341.Berlin:Springer-Verlag.8Mayer,A.and Sykes,A.(1991).Statistical Methods for the Analysis of Soft-ware Metrics Data.In Proceedings of the2nd Annual Oregon Workshop on Software Metrics Data,Portland.McCabe,T.A.(1976).A Software Complexity Measure.IEEE Trans.Soft-ware Engineering,SE-2,308-320.Meinhold,R.J.and Singpurwalla,N.D.(1983).Bayesian Analysis of a Com-monly Used Model for Describing Software Failures.The Statistician, 32,168-173.Musa,J.D.(1993).Operational pro…le in software-reliability engineering.IEEE Transaction on Software,10,14-32.Musa,J.D.and Okumoto,K.(1987).A logarithmic Poisson execution time model for software reliability measurement.Proceedings of the 7th International Conference on Software Engineering,Orlando,230–237.Schick,G.J.and Wolverton,R.W.(1978).Assessment of Software Relia-bility.In Proceedings in Operations Research,Physica-Verlag,Vienna, 395–422.Singpurwalla,N.D.and Wilson,S.P.(1994).Software reliability modeling.Int.Statist.Rev.,62,3:289–317.Singpurwalla,N.D.and Wilson,S.P.(1999).Statistical Methods in Soft-ware Engineering:Reliability and Risk,New York:Springer-Verlag.Spiegelhalter,D.J.,Thomas,A.and Best,N.G.(1999).WinBugs Version1.2User Manual.MRC Biostatistics Unit,Cambridge University.Wilson,S.P.and Wiper,M.P.(2000).Prior Robustness in some Common Types of Software Reliability Model.In Robust Bayesian Analysis,(D.Ríos Insua and F.Ruggeri eds.)New York:Springer-Verlag.Wiper,M.P.,Brunenberg,L.and Göbbels,M.(1992).Relation between software metrics.In Proceedings of Eurometrics’92;European Confer-ence on Quantitative Evaluation of Software&Systems-Practical and Theoretical Aspects,EC2:Paris,91–99.Wiper,M.P.,Ríos Insua,D.and Hierons,R.M.(1998).Bayesian inference and optimal release times for two software failure models.Revista de la Real Academia de Ciencias Exactas,Fisicas y Naturales(España), 92,323–328.9。