Bayesian Nonparametric Adaptive Control using Gaus

informativecensoring包使用方法Informative censoring包使用方法简介informativecensoring包是一个用于处理有信息删失数据的R 软件包。

它提供了一系列方法和函数，方便我们对有信息删失数据进行建模和统计分析。

方法一：Kaplan-Meier估计•使用kc函数来进行Kaplan-Meier估计。

•输入参数包括观测时间和观测状态，返回生存曲线估计结果。

方法二：Cox比例风险模型•使用coxph函数来拟合Cox比例风险模型。

•输入参数包括观测时间、观测状态和预测变量，返回模型的风险比估计和显著性检验结果。

方法三：多状态模型•使用msm函数来拟合多状态模型。

•输入参数包括观测时间、观测状态和转移矩阵，返回模型的状态过渡概率估计和转移时间的估计。

方法四：Guyot方法•使用guyot函数来进行Guyot方法估计。

•输入参数包括观测时间、观测状态和参考组信息，返回Guyot方法估计的结果。

方法五：Preston方法•使用preston函数来进行Preston方法估计。

•输入参数包括观测时间、观测状态和参考组信息，返回Preston 方法估计的结果。

方法六：Latouche方法•使用latouche函数来进行Latouche方法估计。

•输入参数包括观测时间、观测状态和参考组信息，返回Latouche 方法估计的结果。

结论informativecensoring包提供了多种处理有信息删失数据的方法，包括Kaplan-Meier估计、Cox比例风险模型、多状态模型以及Guyot、Preston和Latouche方法。

通过使用这些方法，我们可以更好地处理有信息删失数据，并进行更准确的统计分析。

方法七：IPW方法•使用ipw函数来进行倾向性权重（IPW）方法估计。

•输入参数包括观测时间、观测状态和倾向性得分，返回使用IPW 方法估计的结果。

方法八：多重失效时间模型（MMRM）•使用mrm函数来进行多重失效时间模型（MMRM）估计。

Construction and Evaluation of Actuarial Models ExamThe Construction and Evaluation of Actuarial Models exam is called Exam C by the SOA and Exam 4 by the CAS. This three-and-a-half hour exam consists of 35 multiple-choice questions. The examination is jointly sponsored and administered by the SOA, CAS, and the Canadian Institute of Actuaries (CIA). The examination is also jointly sponsored by the American Academy of Actuaries (AAA) and the Conference of Consulting Actuaries (CCA).The Construction and Evaluation of Actuarial Models Exam is administered as a computer-based test. For additional details, please refer to Exam Rules. The syllabus for this examination provides an introduction to modeling and covers important actuarial methods that are useful in modeling. A thorough knowledge of calculus, probability, and mathematical statistics is assumed.The candidate will be introduced to a variety of useful frequency and severity models. The candidate will be required to understand the steps involved in the modeling process and how to carry out these steps in solving business problems. The candidate should be able to: 1) analyze data from an application in a business context; 2) determine a suitable model including parameter values; and 3) provide measures of confidence for decisions based upon the model. The candidate will be introduced to a variety of tools for the calibration and evaluation of the models.A variety of tables is available below for the candidate and will be provided to the candidate at the examination. These include values for the standard normal distribution, chi-square distribution, and abridged inventories of discrete and continuous probability distributions. Candidates will not be allowed to bring copies of the tables into the examination room. A preview of the CBT layout of the Exam C/4 tables in HTML is available online since the tables have been divided into five sections for viewing purposes.Check the Updates section on this exam's home page for any changes to the exam or syllabus. In the learning outcomes, weights have been provided to indicate the relative emphasis on different sections. The ranges of weights shown are intended to apply to the large majority of exams administered. On occasion, the weights of topics on an individual exam may fall outside the published range. Candidates should also recognize that some questions may cover multiple learning outcomes.Each multiple-choice problem includes five answer choices identified by the letters A, B, C, D, and E, only one of which is correct. Candidates must indicate responses to each question on the computer.As part of the computer-based testing process, a few pilot questions will be randomly placed in the exam (paper and pencil and computer-based forms). These pilot questions are included to judge their effectiveness for future exams, but they will NOT be used in the scoring of this exam.1 All other questions will be considered in the scoring. All unanswered questions are scored incorrect. Therefore, candidates should answer every question on the exam. There is no set requirement for the distribution of correct answers for the SOA/CAS/CIA multiple-choice preliminary examinations. It is possible that a particular answer choice could appear many times on an examination or not at all. Candidates are advised to answer each question to the best of their ability, independently from how they have answered other questions on the examination.Because the CBT exam will be offered over a period of a few days, each candidate will receive a test form composed of questions selected from a pool of questions. Statistical scaling methods are used to ensure within reasonable and practical limits that, during the same testing period of a few days, all forms of the test are comparable in content and passing criteria. The methodology that has been adopted is used by many credentialing programs that give multiple forms of an exam.LEARNING OUTCOMESThe candidate is expected to be familiar with survival, severity, frequency and aggregate models, and use statistical methods to estimate parameters of such models given sample data. The candidate is further expected to identify steps in the modeling process, understand the underlying assumptions implicit in each family of models, recognize which assumptions are applicable in a given business application, and appropriately adjust the models for impact of insurance coverage modifications.Specifically, the candidate is expected to be able to perform the tasks listed below. Items in italic font are additions or replacements with respect to the June 2013 syllabus.Sections A–E have a combined weight of 15-20%.A. Severity Models1. Calculate the basic distributional quantities:a) momentsb) Percentilesc) Generating functions2. Describe how changes in parameters affect the distribution.3. Recognize classes of distributions and their relationships.4. Apply the following techniques for creating new families of distributions:a) Multiplication by a constant1 Beginning with the October 2013 examination there are some revised learning objectives and readings. Items covering them may appear as scored or pilot items in the same manner as items on continuing topics.b) Raising to a powerc) Exponentiation,d) Mixing5. Identify the applications in which each distribution is used and reasons why.6. Apply the distribution to an application, given the parameters.7. Calculate various measures of tail weight and interpret the results to compare the tailweights.8. Identify and describe two extreme value distributions.B. Frequency ModelsFor the Poisson, Mixed Poisson, Binomial, Negative Binomial, Geometric distribution and mixtures thereof:1. Describe how changes in parameters affect the distribution,2. Calculate moments,3. Identify the applications for which each distribution is used and reasons why,4. Apply the distribution to an application given the parameters.5. Apply the zero-truncated or zero-modified distribution to an application given theparametersC. Aggregate Models1. Compute relevant parameters and statistics for collective risk models.2. Evaluate compound models for aggregate claims.3. Compute aggregate claims distributions.D. For severity, frequency and aggregate models1. Evaluate the impacts of coverage modifications:a) Deductiblesb) Limitsc) Coinsurance2. Calculate Loss Elimination Ratios.3. Evaluate effects of inflation on losses.E. Risk Measures1. Calculate VaR, and TVaR and explain their use and limitations.Sections F and G have a combined weight of 20-25%.F. Construction of Empirical Models1. Estimate failure time and loss distributions using:a) Kaplan-Meier estimatorb) Nelson-Åalen estimatorc) Kernel density estimators2. Estimate the variance of estimators and confidence intervals for failure time and lossdistributions.3. Apply the following concepts in estimating failure time and loss distribution:a) Unbiasednessb) Consistencyc) Mean squared errorG. Estimation of decrement probabilities from large samples1. Estimate decrement probabilities using both parametric and nonparametric approachesfor both individual and interval data2. Approximate the variance of the estimatorsH. Construction and Selection of Parametric Models (25-30%)1. Estimate the parameters of failure time and loss distributions using:a) Maximum likelihoodb) Method of momentsc) Percentile matchingd) Bayesian procedures2. Estimate the parameters of failure time and loss distributions with censored and/ortruncated data using maximum likelihood.3. Estimate the variance of estimators and the confidence intervals for the parameters andfunctions of parameters of failure time and loss distributions.4. Apply the following concepts in estimating failure time and loss distributions:a) Unbiasednessb) Asymptotic unbiasednessc) Consistencyd) Mean squared errore) Uniform minimum variance estimator5. Determine the acceptability of a fitted model and/or compare models using:a) Graphical proceduresb) Kolmogorov-Smirnov testc) Anderson-Darling testd) Chi-square goodness-of-fit teste) Likelihood ratio testf) Schwarz Bayesian CriterionI. Credibility (20-25%)1. Apply limited fluctuation (classical) credibility including criteria for both full and partialcredibility.2. Perform Bayesian analysis using both discrete and continuous models.3. Apply Bühlmann and Bühlmann-Straub models and understand the relationship of theseto the Bayesian model.4. Apply conjugate priors in Bayesian analysis and in particular the Poisson-gamma model.5. Apply empirical Bayesian methods in the nonparametric and semiparametric cases. J. Simulation (5-10%)1. Simulate both discrete and continuous random variables using the inversion method.2. Simulate from discrete mixtures, decrement tables, the (a,b,0) class, and the normal andlognormal distributions using methods designed for those distributions3. Estimate the number of simulations needed to obtain an estimate with a given error anda given degree of confidence.4. Use simulation to determine the p-value for a hypothesis test.5. Use the bootstrap method to estimate the mean squared error of an estimator.6. Apply simulation methods within the context of actuarial models.Reading Selections for learning outcomes A through H and J:Text• Loss Models: From Data to Decisions, (Fourth Edition), 2012, by Klugman, S.A., Panjer,H.H. and Willmot, G.E.Chapter 3Chapter 4Chapter 5Chapter 6Chapter 8Chapter 9, Sections 9.1–9.7 (excluding 9.6.1), Sections 9.8.1–9.8.2Chapter 10Chapter 11Chapter 12Chapter 13Chapter 14, Sections 14.1 – 14.4 and 14.6Chapter 15Chapter 16Chapter 20Reading Options for learning outcome I (Credibility) will be:Option ACandidates may prepare for this exam using either the Third or Fourth Editions.• Loss Models: From Data to Decisions, (Fourth Edition), 2012, by Klugman, S.A., Panjer,H.H., and Willmot, G.E.Chapter 17, Sections 17.2 – 17.7Chapter 18Chapter 19Option B•Foundations of Casualty Actuarial Science (Fourth Edition), 2001, Casualty Actuarial SocietyChapter 8, Section 1 (background only) Sections 2–5•Topics in Credibility by Dean, C.G.Option C• Introduction to Credibility Theory (Fourth Edition), 2010, Herzog, T.N.Chapters 1-3 (background only)Chapters 4–8Chapter 9 (background only)Other ResourcesTables for Exam C/Exam 4Preview of the CBT layout of the Exam C/4 tables in HTMLAll released exam papers since 2000, can be found at:Past Exam Questions and SolutionsExam C/4 Sample Questions and Solutions. These documents will be updated to reflect the October 2013 syllabus changes. The cover page of each document will indicate any updates that have been made.。

- 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。

统计学习理论的本质：英中文术语对照表来源：张学工译, VN Vapnik原著, 统计学习理论的本质, 清华大学出版社, 2000使用范围：南京师范大学计算机科学与技术学院研究生。

声明：任何人在其出版物使用或者上载到互连网都必须得到译者及出版社的许可。

AdaBoost algorithm （AdaBoost(自举)算法）163admissible structure （容许结构） 95algorithmic complexity （算法复杂度） 10annealed entropy （退火熵） 55ANOVA decomposition （ANOVA分解） 199a posteriori information （后验信息） 120a priori information （先验信息） 120approximately defined operator （近似定义的算子） 230 approximation rate （逼近速率） 98artificial intelligence （人工智能） 13axioms of probability theory （概率理论的公理） 60back propagation method （后向传播方法） 126basic problem of probability theory （概率论的基本问题） 62basic problem of statistics （统计学的基本问题） 63Bayesian approach （贝叶斯方法） 119Bayesian inference （贝叶斯推理） 34bound on the distance to the smallest risk （与最小风险的距离的界） 77 bound on the values of achieved risk （所得风险值的界） 77bounds on generalization ability of a learning machine （学习机器推广能力的界） 76canonical separating hyperplanes （标准分类超平面） 132capacity control problem （容量控制问题） 116cause-effect relation （因果关系） 9choosing the best sparse algebraic polynomial （选择最佳稀疏多项式）117choosing the degree of polynomial （选择多项式阶数） 116 classification error （分类错误） 19codebook （码本） 106complete (Popper's) nonfalsifiability （完全(波普)不可证伪性） 52 compression coefficient （压缩系数） 107consistency of inference （推理的一致性） 36constructive distribution-independent bound on the rate of convergence （构造性的不依赖于分布的收敛速度界） 69convolution of inner production （内积回旋） 140criterion of nonfalsifiability （不可证伪性判据） 47data smoothing problem （数据平滑问题） 209decision-making problem （决策选择问题） 296decision trees （决策树） 7deductive inference （演绎推理） 47density estimation problem （密度估计问题）：parametric(Fisher-Wald) setting（参数化(Fisher-Wald)表示） 20nonparametric setting （非参数表示） 28discrepancy （差异） 18discriminant analysis （判别分析） 24discriminant function （判别函数） 25distribution-dependent bound on the rate of convergence （依赖于分布的收敛速度界） 69distribution-independent bound on the rate of convergence （不依赖于分布的收敛速度界） 69ΔΔ-margin separating hyperplane （间隔分类超平面） 132 empirical distribution function （经验分布函数） 28empirical processes （经验过程） 40empirical risk functional （经验风险泛函） 20empirical risk minimization inductive principle （经验风险最小化归纳原则） 20ensemble of support vector machines （支持向量机的组合） 163 entropy of the set of functions （函数集的熵） 42entropy on the set of indicator functions （指示函数集的熵） 42 equivalence classes （等价类） 292estimation of the values of a function at the given points （估计函数在给定点上的值） 292expert systems （专家系统） 7ε-insensitivity （ε不敏感性） 181ε-insensitive loss function （ε不敏感损失函数） 181feature selection problem （特征选择问题） 118function approximation （函数逼近） 98function estimation model （函数估计模型） 17Gaussian （高斯函数） 26generalized Glivenko-Cantelli problem （广义Glivenko-Cantelli问题）66generalized growth function （广义生长函数） 85generator random vectors （随机向量产生器） 17Glivenko-Cantelli problem （Glivenko-Cantelli问题） 66growth function （生长函数） 55Hamming distance （汉明距离） 104handwritten digit recognition （手写数字识别） 146hard threshold vicinity function （硬限邻域函数） 103hard vicinity function （硬领域函数） 269hidden Markov models （隐马尔可夫模型） 7hidden units （隐结点） 101Huber loss function （Huber损失函数） 183ill-posed problems （不适定问题）： 9solution by variation method （变分方法解） 236solution by residual method （残差方法解） 236solution by quasi-solution method （拟解方法解） 236 independent trials （独立试验） 62inductive inference （归纳推理） 50inner product in Hilbert space （希尔伯特空间中的内积） 140 integral equations （积分方程）：solution for exact determined equations （精确确定的方程的解）237solution for approximately determined equations （近似确定的方程的解） 237kernel function （核函数） 27Kolmogorov-Smirnov distribution （Kolmogorov-Smirnov分布） 87 Kulback-Leibler distance （Kulback-Leibler距离） 32Kuhn-Tücker conditions （库恩-塔克条件） 134Lagrangian multiplier （拉格朗日乘子） 133Lagrangian （拉格朗日函数） 133Laplacian （拉普拉斯函数） 277law of large number in the functional space （泛函空间中的大数定律）41law of large numbers （大数定律） 39law of large numbers in vector space （向量空间中的大数定律） 41 Lie derivatives （Lie导数） 20learning matrices （学习矩阵） 7least-squares method （最小二乘方法） 21least-modulo method （最小模方法） 182linear discriminant function （学习判别函数） 31linearly nonseparable case （线性不可分情况） 135local approximation （局部逼近） 104local risk minimization （局部风险最小化） 103locality parameter （局部性参数） 103loss-function （损失函数）：for AdaBoost algorithm （AdaBoost算法的损失函数） 163for density estimation （密度估计的损失函数） 21for logistic regression （逻辑回归的损失函数） 156for pattern recognition （模式识别的损失函数） 21for regression estimation （回归估计的损失函数） 21 madaline（Madaline自适应学习机） 7main principle for small sample size problems （小样本数问题的基本原则） 28maximal margin hyperplane （最大间隔超平面） 131maximum likehood method （最大似然方法） 24McCulloch-Pitts neuron model （McCulloch-Pitts神经元模型） 2 measurements with the additive noise （加性噪声下的测量） 25 metric ε-entropy （ε熵度量） 44minimum description length principle （最小描述长度原则） 104 mixture of normal densities （正态密度的组合） 26National Institute of Standard and Technology (NIST) digit database （美国国家标准技术研究所(NIST)数字数据库） 173neural networks （神经网络） 126non-trivially consistent inference （非平凡一致推理） 36 nonparametric density estimation （非参数密度估计） 27normal discriminant function （正态判别函数） 31one-sided empirical process （单边经验过程） 40optimal separating hyperplane （最优分类超平面） 131overfitting phenomenon （过学习现象） 14parametric methods of density estimation （密度估计的参数方法） 24 partial nonfalsifiability （部分不可证伪性） 51Parzen's windows method （Parzen窗方法） 27pattern recognition problem （模式识别问题） 19perceptron （感知器） 1perceptron's stopping rule （感知器迭代终止规则） 6polynomial approximation of regression （回归的多项式逼近） 116 polynomial machine （多项式机器） 143potential nonfalsifiability （潜在不可证伪性） 53probability measure （概率测度） 59probably approximately correct (PAC) model （可能近似正确(PAC)模型） 13problem of demarcation （区分问题） 49pseudo-dimension （伪维） 90quadratic programming problem （二次规划问题） 133quantization of parameters （参数的量化） 110quasi-solution （拟解） 112radial basis function machine （径向基函数机器） 144random entropy （随机熵） 42radnom string （随机串） 10randomness concept （随机性概念） 10regression estimation problem （回归估计问题） 19regression function （回归函数） 19regularization theory （正则化理论） 9regularized functional （正则化泛函） 9reproducing kernel Hilbert space （再生核希尔伯特空间） 244 residual principle （残差原则） 236rigorous (distribution-dependent) bounds （严格(依赖于分布的)界） 85 risk functional （风险泛函） 18risk minimization from empirical data problem （基于经验数据最小化风险的问题） 20robust estimators （鲁棒估计） 26robust regression （鲁棒回归） 26Rosenblatt's algorithm （Rosenblatt算法） 5set of indicators （指示器集合） 73set of unbounded functions （无界函数集合） 77σ-algebra （σ代数） 60sigmoid function （S型(sigmoid)函数） 125small samples size （小样本数） 93smoothing kernel （平滑核） 100smoothness of functions （函数的平滑性） 100soft threshold vicinity function （软阈值领域函数） 103soft vicinity function （软领域函数） 269soft-margin separating hyperplane （软间隔分类超平面） 135spline function （样条函数）：with a finite number of nodes （有限结点的样条函数） 194with an infinite number of nodes （无穷多结点的样条函数） 195 stochastic approximation stopping rule （随机逼近终止规则） 34 stochastic ill-posed problems （随机不适定问题） 113strong mode estimating a probability measure （强方式概率度量估计）63structural risk minimization principle （结构风险最小化原则） 94 structure （结构） 94structure of growth function （生长函数的结构） 79supervisor （训练器） 17support vector machines （支持向量机） 137support vectors （支持向量） 134support vector ANOVA decomposition （支持向量ANOVA分解） 199 SVM n approximation of the logistic regression （逻辑回归的SVM n逼近） 155SVM density estimator （SVM密度估计） 246SVM conditional probability estimator （SVM条件概率估计） 257 tails of distribution （分布的尾部） 78tangent distance （切距） 149training set （训练集） 18transductive inference （转导推理） 293Turing-Church thesis （Turing-Church理论） 177two layer neural networks machine （两层神经网络机器） 145two-sided empirical process （双边经验过程） 40U.S. Postal Service digit database （美国邮政数字数据库） 173 uniform one-sided convergence （一致单边收敛） 39uniform two-sided convergence （一致双边收敛） 39VC dimension of a set of indictor functions （指示函数集的VC维） 79 VC dimension of a set of real functions （实函数集的VC维） 81VC entropy （VC熵） 44VC subgraph （VC子图） 90vicinal risk minimization method（领域风险最小化） 268vicinity kernel（领域核）：273one-vicinal kernel （单领域核） 273two-vicinal kernel （双领域核） 273VRM method （VRM方法）：for pattern recognition （模式识别的VRM方法） 273for regression estimation （回归估计的VRM方法） 282for density estimation （密度估计的VRM方法） 284for conditional probability estimation （条件概率估计的VRM方法） 285for conditional density estimation （条件密度估计的VRM方法）286weak mode estimating a probability measure （弱方式概率度量估计）63weight decay procedure （权值衰减过程） 102。