贝叶斯模型：推断与预测的深入解析

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"《贝叶斯建模、推断与预测》是一本由David Draper撰写的专业书籍，主要探讨了在统计学和机器学习领域中的核心概念——贝叶斯方法。作者是美国加利福尼亚大学圣克鲁兹分校应用数学与统计系的教授，他的电子邮件地址为draper@ams.ucsc.edu，可以通过该链接获取更多资料。本书于2005年12月发布，第6版初稿，强调了内容的实用性并欢迎读者提供反馈。贝叶斯模型是本书的核心，它是一种基于贝叶斯定理的概率论框架，用于处理不确定性问题，特别适用于处理缺失数据和模型选择。贝叶斯定理是概率论中的基石，它阐述了在已知某些先验知识的情况下，如何更新对未知事件的概率估计。通过这个定理，我们可以将新观测到的数据与先前的假设相结合，从而得到更精确的推断结果。预测分析在本书中占据重要地位，利用贝叶斯模型进行预测时，可以处理复杂的动态系统和时间序列数据，这使得模型具有强大的适应性和灵活性。贝叶斯决策理论则进一步扩展了这一主题，它指导我们在不确定性的环境中做出最佳决策，通过对不同选项的后验概率进行比较，确定最有可能带来最大收益的选择。书中还涉及到了技术层面的细节，包括如何用PostScript-based phototypesetter进行排版，以及如何使用R语言（R Project，2005）生成图形，并直接嵌入到文档中。此外，作者使用LATEX语言（Lamport, 1994）进行文本格式化，展示了其在科技出版中的应用。本书旨在为读者提供一个全面的贝叶斯方法基础，适合对统计学、机器学习、数据科学感兴趣的读者深入理解这些关键概念，并将其应用于实际问题解决中。通过阅读和实践，读者不仅可以掌握贝叶斯模型的基本原理，还能学习到如何进行有效的模型推断和预测，以及如何在决策制定中充分利用这些统计工具。"

6 David Draper

arrived; this is updated multiplicatively on the probability scale by the like-

lihood P (data|unknown), and renormalized so that total probability remains

Writing the Theorem both for A and (not A) and combining gives a

(perhaps even more) useful version:

Bayes’ Theorem in odds form

P (A|data)

P (not A|data)

posterior

odds

P (A)

P (not A)

prior

odds

P (data|A)

P (data|not A)

Bayes

factor

Another name for the Bayes factor is the likelihood ratio, since this factor

measures the relative plausibility of the data given A and (not A).

Applying this to the HIV example requires additional information about

ELISA obtained by screening the blood of people with known HIV status:

sensitivity = P (ELISA positive|HIV positive) and

speciﬁcity = P (ELISA negative|HIV negative)

In practice ELISA’s operating characteristics are (or at least seem) rather

good—sensitivity about 0.95, speciﬁcity about 0.98—so you might well ex-

pect that

P (this patient HIV positive|ELISA positive)

would be close to 1.

Here the updating produces a surprising result: the Bayes factor comes

out

B =

sensitivity

1 − speciﬁcity

0.95

0.02

= 47.5,

which sounds like strong evidence that this patient is HIV positive, but

the prior odds are quite a bit stronger the other way (

P (A)

1−P (A)

= 99 to 1

against HIV) leading to posterior odds of

47.5

= 2.08 against HIV, i.e.,

P (HIV positive|data) =

1+odds

293

= 0.32 (!).

The reason for this is that ELISA was designed to have a vastly better

false negative rate—P (HIV positive| ELISA negative) =

9707

= 0.00052

Bayesian Modeling, Inference and Prediction 7

1 in 1941—than false positive rate—P (HIV negative| ELISA positive) =

198

293

= 0.68

= 2 in 3. This in turn is because ELISA’s developers judged that

it’s far worse to tell somebody who’s HIV positive that they’re not than the

other way around (this is a reasonable position when using ELISA for, e.g.,

blood bank screening, which is one of the main uses for the test). This false

positive rate would make widespread screening for HIV based only on ELISA

a truly bad idea.

Formalizing the consequences of the two types of error in diagnostic

screening would require quantifying the misclassiﬁcation costs, which shifts

the focus from (scientiﬁc) inference (the acquisition of knowledge for its own

sake: Is this patient really HIV-positive?) to decision-making (putting that

knowledge to work to answer a public policy or business question: What use

of ELISA and Western Blot would yield the optimal screening strategy?).

1.3 Bayesian decision theory

Axiomatic approaches to rational decision-making date back to Ramsay

(1926), with von Neumann and Morgenstern (1944) and Savage (1954) also

making major contributions. The ingredients of a general decision problem

(e.g., Bernardo and Smith, 1994) include

• A set {a

, i ∈ I} of available actions, one of which I will choose;

• For each action a

, a set {E

, j ∈ J} of uncertain outcomes describ-

ing what will happen if I choose action a

;

• A set {c

, j ∈ J} of consequences corresponding to the outcomes

, j ∈ J}; and

• A preference relation ≤, expressing my preferences between pairs of

available actions (a

≤ a

means “a

is not preferred by me to a

”).

Deﬁne a

∼ a

(“a

and a

are equivalent” to me) iﬀ a

≤ a

and

≤ a

This preference relation induces a qualitative ordering of the uncertain out-

comes (E ≤ F means “E is not more likely than F ”), because if I compare

two dichotomized possible actions, involving the same consequences and dif-

fering only in their uncertain outcomes, the fact that I prefer one action to

8 David Draper

another means that I must judge it more likely that if I take that action the

preferred consequence will result.

Within this framework I have to make further assumptions—the coher-

ence axioms—to ensure that my actions are internally consistent. Informally

(see Bernardo and Smith, 1994, for the formalism) these are:

• An axiom insisting that I be willing to express preferences between

simple dichotomized possible actions ({a, not a});

• A transitivity axiom in which (for all actions a, a

, a

) a ≤ a, and

if a

≤ a

and a

≤ a

then a

≤ a

; and

• An axiom based on the sure-thing principle: if, in two situations,

no matter how the ﬁrst comes out the corresponding outcome in the

second is preferable, then I should prefer the second situation overall.

This puts ≤ on a sound footing for qualitative uncertainty assessment,

but does not yet imply how to quantify—it’s like being able to say that one

thing weighs less than another but not to say by how much. To go further

requires a fourth assumption, analogous to the existence of a set of reference

standards (e.g., an oﬃcial kg weight, half-kg, and so on) and the ability to

make arbitrarily precise comparisons with these standards:

• An axiom guaranteeing that for each outcome E there exists a stan-

dard outcome S (e.g., “idealized coin lands heads”) such that E ∼ S.

This framework implies the existence and uniqueness of a (personal) prob-

ability P

(abbreviated P ), mapping from outcomes E to [0,1] and corre-

sponding to the judgments in my deﬁnition of ≤, and a utility function U

(large values preferred, say), mapping from consequences c to R and quanti-

fying my preferences.

This has all been rather abstract. Two concrete results arising from this

framework may make its implications clearer:

• Bayes’ original deﬁnition of personal probability is helpful in thinking

about how to quantify uncertainty. Pretending that consequences are

monetary (e.g., US$), for me to say that P

(E) = p for some uncertain

outcome E whose truth value will be known in the future is to say that

I’m indiﬀerent between (a) receiving $p · m for sure (for some (small)

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