探索Bayesian方法：R语言实现与LearnBayes包

R语言

贝叶斯

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"《Bayesian Computation with R》是由Jim Albert编写的，旨在介绍贝叶斯方法的入门书籍。这本书适合统计学初学者和希望了解更多R语言及贝叶斯方法的专业人士。书中提到的LearnBayes包，由作者编写，可以在CRAN网站上找到，包含了书中所有描述的R函数。此外，本书是UseR!系列的一部分，该系列还包括其他与R语言和数据分析相关的著作。" 正文：《Bayesian Computation with R》是一本深入浅出地讲解贝叶斯计算的书籍，特别关注如何使用R语言来实施贝叶斯统计分析。贝叶斯统计是一种概率统计方法，它允许我们基于先验知识更新对事件概率的理解，随着新数据的引入，这些先验知识会不断演化为后验概率。这种方法在许多科学领域，如生物统计、机器学习和数据科学中都得到了广泛应用。本书的主要目标是教育读者理解贝叶斯推理的基本原理，并提供实践工具，使他们能够使用R语言进行贝叶斯建模。R语言是一个强大的开源统计分析环境，因其丰富的统计功能和易于编程而受到统计学家和数据科学家的青睐。通过LearnBayes包，读者可以方便地实现书中介绍的各种贝叶斯模型，从而加深对理论知识的理解。书中可能涵盖了以下几个核心知识点： 1. **贝叶斯公式**：解释了贝叶斯定理的数学基础，它是将先验概率与似然性相结合以计算后验概率的关键工具。 2. **贝叶斯模型构建**：介绍如何设置先验分布，选择合适的模型结构，以及如何用R语言实现这些模型。 3. **马尔可夫链蒙特卡洛（MCMC）**：作为求解复杂的后验分布的常用方法，MCMC如Metropolis-Hastings算法或Gibbs采样，会在书中有所阐述。 4. **数据模拟与诊断**：讨论如何使用R进行模拟研究，以验证模型的正确性和合理性，以及如何评估MCMC的收敛性。 5. **应用案例**：可能会包含实际数据集的应用示例，展示如何将贝叶斯方法应用于解决现实世界的问题。 6. **R函数详解**：LearnBayes包中的每个函数及其用途都会被详细介绍，帮助读者在实践中掌握这些工具。 7. **结果解释与可视化**：介绍如何解释贝叶斯模型的结果，并利用R进行有效的数据可视化。这本书不仅对统计学学生有指导价值，也对统计实践者非常有用，因为它提供了理论知识与实践经验的完美结合。通过阅读和实践，读者不仅可以理解贝叶斯方法的核心概念，还能熟练掌握使用R进行贝叶斯分析的技能。

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1.2 Exploring a Student Dataset 5

hours.of.sleep

Frequency

24681012

0 50 100 150

Fig. 1.2. Histogram of the hours of sleep of the sta tistics students.

Gender=="female"

is a logical statement that will be TRUE if Gender is “female”; otherwise it will

be FALSE. The expression

Haircut[condition]

will produce a subset of Haircut according to when the condition is TRUE.

So the statement

> female.Haircut=Haircut[Gender=="female"]

will select the haircut prices only for the female students and store the prices

into the variable female.Haircut. Similarly, we use the logical operator to

store the male haircut prices into the variable male.Haircut.

> male.Haircut=Haircut[Gender=="male"]

By use of the summary command, we summarize the haircut prices of the

women and the men.

> summary(female.Haircut)

Min. 1st Qu. Median Mean 3rd Qu. Max. NA’s

0.00 15.00 25.00 34.08 45.00 180.00 19.00

1.3 Exploring the Robustness of the t Statistic 9

: µ

= µ

Let

X and

Y denote the sample means of the xsandys and let s

and s

denote the respective standard deviations. The standard test of this hypothesis

is based on the t statistic

T =

X −



1/m +1/n

where s

is the pooled stan dar d deviation



(m −1)s

+(n − 1)s

m + n − 2

Under the hypothesis H

, the test statistic T has a t distribution with m+n−2

degrees of freedom when

• both the xsandys are i nd ependent random samples from normal distrib-

utions

• the standard deviations of the x and y popul ations, σ

and σ

, are equal

Suppose the level of signiﬁcance of the test is set at α. Then one will reject

H when

|T |≥t

n+m−2,α/2

where t

df , α

is the (1 − α) quantile of a t random variable with df degrees of

freedom.

If the underlying assumptions of normal populations and equal variances

hold, then the level of signiﬁcance of the t-test will be the stated level of

α. But in practice, many people use the t statistic to compare two samples

even when the underlying assumptions are in doubt. So an interesting prob-

lem is to investigate the robustness or sensitivity of this popular test statistic

with respect to changes in the assumptions. If the stated signiﬁcance level is

α = .10 and the populations are skewed or have heavy tails, what will be the

true signiﬁcance level? If the assumption of equal variances is violated and

there are signiﬁcant diﬀerences in the spreads of the two populations, what is

the true signiﬁcance level? One can answer these questions t hrou gh a Monte

Carlo simulation study. R is a very suitable platform for writing a simulation

algorithm. One can generate random samples from a wide variety of proba-

bility distributions and R has an extensive set of data analysis capabilities

for summarizing and graphing the simulation output. Here we illustrate the

construction of a simple R function to address the r obu stness of the t statistic.

1.3.2 Writing a Function to Compute the t Statistic

To begin, we generate some random data for the samples of xsandys. We

simulate a sample of 10 observations from a normal distribution with mean

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