入门贝叶斯统计：Springer《第一堂贝叶斯统计方法课》

Bayesian

Statistics

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"《A First Course in Bayesian Statistical Methods》是一本由Peter D. Hoff编写的入门级贝叶斯统计学教材，旨在为初学者提供一个理解贝叶斯方法的平易近人的途径。书中的公式相对简单，适合统计学和数据分析领域的初学者。该书属于SpringerTexts in Statistics系列，由G. Casella、S. Fienberg和I. Olkin等知名统计学家编辑。读者可以通过Springer出版社的网站获取该系列的其他书籍信息。" 在《A First Course in Bayesian Statistical Methods》中，作者Peter D. Hoff详细介绍了贝叶斯统计的基本概念和方法。贝叶斯统计是一种统计推理方式，它以概率理论为基础，通过先验概率和观测数据来更新我们对未知参数的信念。这本书的核心内容可能包括以下几个方面： 1. **贝叶斯定理**：首先，书中会详细解释贝叶斯定理的数学形式和其背后的直观理解，它是贝叶斯分析的基础，允许我们将先验信息与观察数据相结合。 2. **先验分布**：讨论如何选择和使用不同的先验分布，如均匀分布、正态分布、贝塔分布等，并探讨它们在不同情况下的适用性。 3. **后验分布**：介绍如何根据观测数据计算后验分布，这是贝叶斯分析的关键步骤，它提供了在考虑到所有证据后对模型参数的最新估计。 4. **马尔可夫链蒙特卡洛(MCMC)**：由于复杂的后验分布可能无法直接计算，书可能会涵盖MCMC方法，如Metropolis-Hastings算法或Gibbs采样，这些是求解高维后验分布的有效工具。 5. **贝叶斯决策理论**：书中可能涉及如何利用后验分布进行决策制定，包括风险函数和决策规则的概念。 6. **应用实例**：通过实际案例，比如医学诊断、社会科学、工程问题等，展示贝叶斯方法在解决实际问题中的应用。 7. **贝叶斯模型比较**：介绍贝叶斯因素或其他方法用于比较不同模型的合理性，帮助选择最佳模型。 8. **贝叶斯网络**：可能会介绍贝叶斯网络的概念，这是一种图形模型，用于表示变量之间的条件依赖关系。通过阅读本书，读者将能够掌握贝叶斯统计的基本原理，学会如何在实际问题中运用这些原理进行数据分析和决策。此外，书中可能还包含了练习题和案例研究，帮助读者巩固所学知识并提高解决问题的能力。对于希望进入统计学或数据分析领域的学习者来说，这是一本理想的入门读物。

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8 1 Introduction and examples

a reliable estimate for large sample sizes, as we saw in the e xample it can be

statistically unreliable for small n, in which case it serves more as a summary

of the sample data than as a precise estimate of θ.

If our interest lies more in obtaining an estimate of θ than in summarizing

our sample data, we may want to consider estimators of the form

θ =

n + w

¯y +

n + w

where θ

represents a “best guess” at the true value of θ and w represents a

degree of conﬁdence in the guess. If the sample size is large, then ¯y is a reliable

estimate of θ. The estimator

θ takes advantage of this by having its weights

on ¯y and θ

go to one and zero, respectively, as n increases. As a result, the

statistical properties of ¯y and

θ are essentially the same for large n. However,

for small n the variability of ¯y might be more than our uncertainty about θ

In this case, using

θ allows us to combine the data with prior information to

stabilize our estimation of θ.

These properties of

θ for both large and small n suggest that it is a useful

estimate of θ for a broad range of n. In Section 5.4 we will conﬁrm this by

showing that, under some conditions,

θ outperforms ¯y as an estimator of θ for

all values of n. As we saw in the infection rate example and will see again in

later chapters,

θ can be interpreted as a Bayesian estimator using a certain

class of prior distributions. Even if a particular prior distribution p(θ) does not

exactly reﬂect our prior information, the corresponding posterior distribution

p(θ|y) can still be a useful means of providing stable inference and es timation

for situations in which the sample size is low.

1.2.2 Building a predictive model

In Chapter 9 we will discuss an example in which our task is to build a pre-

dictive model of diabetes progression as a function of 64 baseline explanatory

variables such as age, sex and body m ass index. Here we give a brief synopsis of

that example. We will ﬁrst estimate the parameters in a regression model us-

ing a “training” dataset consisting of measurements from 342 patients. We will

then evaluate the predictive performance of the estimated regression model

using a separate “test” dataset of 100 patients.

Sampling model and parameter space

Letting Y

be the diabetes progression of subject i and x

= (x

i,1

, . . . , x

i,64

)

be the explanatory variables, we will consider linear regression models of the

form

= β

i,1

+ β

i,2

+ ··· + β

i,64

+ σ

The sixty-ﬁve unknown parameters in this model are the vector of regression

coeﬃcients β = (β

, . . . , β

) as well as σ, the standard deviation of the error

term. The parameter space is 64-dimensional Euclidean s pace for β and the

positive real line for σ.

10 1 Introduction and examples

Predictive performance and comparison to non-Bayesian methods

We can evaluate how well this model performs by using it to predict the test

data: Let

Bayes

= E[β|y, X] be the posterior expectation of β, and let X

test

be the 100 ×64 matrix giving the data for the 100 patients in the test dataset.

We can compute a predicted value for each of the 100 observations in the test

set using the equation

test

= X

Bayes

. These predicted values can then be

compared to the actual obse rvations y

test

. A plot of y

test

versus

test

appears

in the ﬁrst panel of Figure 1.4, and indicates how well

Bayes

is able to predict

diabetes progression from the baseline variables.

How does this Bayesian estimate of β compare to a non-Bayesian ap-

proach? The most commonly used estimate of a vector of regression coeﬃ-

cients is the ordinary least squares (OLS) estimate, provided in most if not all

statistical software packages. The OLS regression estimate is the value

ols

β that minimizes the sum of squares of the residuals (SSR) for the observed

data,

SSR(β) =

i=1

− β

)

and is given by the formula

ols

= (X

−1

y. Predictions for the test

data based on this estimate are given by X

ols

and are plotted against the

observed values in the second panel of Figure 1.4. Notice that using

ols

gives a weaker relationship between observed and predicted values than using

Bayes

. This can be quantiﬁed numerically by computing the average squared

prediction error,

test,i

− ˆy

test,i

)

/100, for both sets of predictions. The

prediction error for OLS is 0.67, about 50% higher than the value of 0.45 we

obtain using the Bayesian estimate. In this problem, even though our ad hoc

prior distribution for β only captures the basic structure of our prior beliefs

(namely, that many of the coeﬃcients are likely to be zero), this is enough to

provide a large improvement in predictive performance over the OLS estimate.

The poor performance of the OLS method is due to its inability to recog-

nize when the sample size is too small to accurately estimate the regress ion

coeﬃcients. In such situations, the linear relationship between the values of

y and X in the dataset, quantiﬁed by

ols

, is often an inaccurate represen-

tation of the relationship in the entire population. The standard remedy to

this problem is to ﬁt a “sparse” regression model, in which some or many

of the regression coeﬃcients are set to zero. One method of choosing which

coeﬃcients to set to zero is the Bayesian approach described above. Another

popular method is the “lasso,” introduced by Tibshirani (1996) and studied

extensively by many others. The lasso estimate is the value

lasso

of β that

minimizes SSR(β : λ), a modiﬁed version of the sum of squared residuals:

SSR(β : λ) =

i=1

− x

β)

+ λ

j=1

|β

1.3 Where we are going 11

●

−1 0 1 2

test

●

−1 0 1 2

test

Fig. 1.4. Observed versus predicted diabetes progression values using the Bayes

estimate (left panel) and the OLS estimate (right panel).

In other words, the lasso procedure p e nalizes large values of |β

|. Depending

on the size of λ, this penalty can make some elements of

lasso

equal to

zero. Although the lasso procedure has been motivated by and studied in

a non-Bayesian context, in fact it corresponds to a Bayesian estimate using

a particular prior distribution: The lasso estimate is equal to the posterior

mode of β in which the prior distribution for each β

is a double-exponential

distribution, a probability distribution that has a sharp peak at β

= 0.

1.3 Where we are going

As the above examples indicate, the uses of Bayesian methods are quite broad.

We have seen how the Bayesian approach provides

• models for rational, quantitative learning;

• estimators that work for small and large sample sizes;

• methods for generating statistical procedures in complicated problems.

An understanding of the b e neﬁts and limits of Bayesian methods comes with

experience. In the chapters that follow, we will become familiar with these

methods by applying them to a large number of statistical models and data

analysis examples. After a review of probability in Chapter 2, we will learn

the basics of Bayesian data analysis and computation in the context of some

simple one-parameter statistical models in Chapters 3 and 4. Chapters 5, 6

and 7 discuss Bayesian inference with the normal and multivariate normal

models. While important in their own right, normal models also provide the

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