揭开贝叶斯统计入门书籍的关键洞察

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《贝叶斯统计入门》是一本深入探讨贝叶斯统计理论的教材，作者在教学过程中发现传统的统计理论对于学生来说存在理解难度，而自己对这一理论的理解也并未随着接触加深而有所改善。于是作者转向研究贝叶斯统计，发现其与经典统计结论之间存在实质性差异，而非仅仅是后者的一种特殊形式。本书共分为三个主要章节： 1. 预备知识：首先，作者介绍了概率论的基础概念和贝叶斯定理，通过实例帮助读者理解定理的应用。接着，讨论了随机变量及其基本性质，包括单个随机变量和多个随机变量的关系，以及均值和方差的概念。章节末尾有大量习题，旨在巩固基础知识。 2. 贝叶斯推理与正态分布：章节重点转向贝叶斯方法在正态分布中的应用。作者解释了贝叶斯推理的本质，即如何结合先验信念和观测数据更新概率。内容涵盖了正态分布的先验假设和似然函数，处理多个正态观察结果的情况，以及高密度区域的分析。此外，作者还探讨了局部均匀先验、正态方差的估计、充足性原则以及共轭分布的重要性，比如指数家族，特别是当正态分布的均值和方差都未知时的共轭联合先验。 3. 其他常见分布：这部分涉及更为广泛的分布，如二项分布及其参考先验，杰弗里斯规则的应用，泊松分布，均匀分布，以及它们的参考先验。书中还涉及两个著名的概率问题：truncated car问题和首位数字问题，讨论了不变先验的概念。章节末尾同样配有练习题，以加深读者对这些理论的实际操作能力。通过阅读这本书，读者将不仅掌握贝叶斯统计的基本原理，还能理解如何将其应用于各种实际情境，特别是在正态分布和其他常见分布的参数估计中。同时，作者对不同先验选择和贝叶斯方法与经典统计方法之间差异的深入剖析，有助于读者开阔视野，理解统计理论的多样性。书中的例子和习题设计旨在帮助读者从理论到实践，提升数据分析和决策制定的能力。

We sometimes need a notation for intervals on the real line, namely

where a and b are real numbers or or .

1.1.2 Axioms for probability

In the study of probability and statistics, we refer to as complete a description of the situation as we

need in a particular context as an elementary event.

Thus, if we are concerned with the tossing of a red die and a blue die, then a typical elementary

event is ‘red three, blue five’, or if we are concerned with the numbers of Labour and Conservative

MPs in the next parliament, a typical elementary event is ‘Labour 350, Conservative 250’. Often,

however, we want to talk about one aspect of the situation. Thus, in the case of the first example, we

might be interested in whether or not we get a red three, which possibility includes ‘red three, blue

one’, ‘red three, blue two’, etc. Similarly, in the other example, we could be interested in whether

there is a Labour majority of at least 100, which can also be analyzed into elementary events. With

this in mind, an event is defined as a set of elementary events (this has the slightly curious

consequence that, if you are very pedantic, an elementary event is not an event since it is an element

rather than a set). We find it useful to say that one event E implies another event F if E is contained in

F. Sometimes it is useful to generalize this by saying that, given H, E implies F if EH is contained in

F. For example, given a red three has been thrown, throwing a blue three implies throwing an even

total.

Note that the definition of an elementary event depends on the context. If we were never going to

consider the blue die, then we could perfectly well treat events such as ‘red three’ as elementary

events. In a particular context, the elementary events in terms of which it is sensible to work are

usually clear enough.

Events are referred to above as possible future occurrences, but they can also describe present

circumstances, known or unknown. Indeed, the relationship which probability attempts to describe is

one between what you currently know and something else about which you are uncertain, both of them

being referred to as events. In other words, for at least some pairs of events E and H there is a

number defined which is called the probability of the event E given the hypothesis H. I might,

for example, talk of the probability of the event E that I throw a red three given the hypothesis H that I

have rolled two fair dice once, or the probability of the event E of a Labour majority of at least 100

given the hypothesis H which consists of my knowledge of the political situation to date. Note that in

this context, the term ‘hypothesis’ can be applied to a large class of events, although later on we will

find that in statistical arguments, we are usually concerned with hypotheses which are more like the

hypotheses in the ordinary meaning of the word.

Various attempts have been made to define the notion of probability. Many early writers claimed

that was m/n where there were n symmetrical and so equally likely possibilities given H of

which m resulted in the occurrence of E. Others have argued that should be taken as the long

run frequency with which E happens when H holds. These notions can help your intuition in some

cases, but I think they are impossible to turn into precise, rigourous definitions. The difficulty with the

first lies in finding genuinely ‘symmetrical’ possibilities – for example, real dice are only

approximately symmetrical. In any case, there is a danger of circularity in the definitions of symmetry

and probability. The difficulty with the second is that we never know how long we have to go on

trying before we are within, say, 1% of the true value of the probability. Of course, we may be able to

give a value for the number of trials we need to be within 1% of the true value with, say, probability

0.99, but this is leading to another vicious circle of definitions. Another difficulty is that sometimes

we talk of the probability of events (e.g. nuclear war in the next 5 years) about which it is hard to

believe in a large numbers of trials, some resulting in ‘success’ and some in ‘failure’. A good, brief

discussion is to be found in Nagel (1939) and a fuller, more up-to-date one in Chatterjee (2003).

It seems to me, and to an increasing number of statisticians, that the only satisfactory way of

thinking of is as a measure of my degree of belief in E given that I know that H is true. It

seems reasonable that this measure should abide by the following axioms:

P1 for all E, H.

P2 = 1 for all H.

P3 when .

By taking in P3 and using P1 and P2, it easily follows that

so that is always between 0 and 1. Also by taking in P3 it follows that

Now intuitive notions about probability always seem to agree that it should be a quantity between 0

and 1 which falls to 0 when we talk of the probability of something we are certain will not happen

and rises to 1 when we are certain it will happen (and we are certain that H is true given H is true).

Further, the additive property in P3 seems highly reasonable – we would, for example, expect the

probability that the red die lands three or four should be the sum of the probability that it lands three

and the probability that it lands four.

Axiom P4 may seem less familiar. It is sometimes written as

although, of course, this form cannot be used if the denominator (and hence the numerator) on the

right-hand side vanishes. To see that it is a reasonable thing to assume, consider the following data on

criminality among the twin brothers or sisters of criminals [quoted in his famous book by Fisher

(1925b)]. The twins were classified according as they had a criminal conviction (C) or not (N) and

according as they were monozygotic (M) (which is more or less the same as identical – we will

return to this in Section 1.2) or dizygotic (D), resulting in the following table:

If we denote by H the knowledge that an individual has been picked at random from this population,

then it seems reasonable to say that

If on the other hand, we consider an individual picked at random from among the twins with a

criminal conviction in the population, we see that

and hence

so that P4 holds in this case. It is easy to see that this relationship does not depend on the particular

numbers that happen to appear in the data.

In many ways, the argument in the preceding paragraph is related to derivations of probabilities

from symmetry considerations, so perhaps it should be stressed that while in certain circumstances

we may believe in symmetries or in equally probable cases, we cannot base a general definition of

probability on such arguments.

It is convenient to use a stronger form of axiom P3 in many contexts, namely,

whenever the (E

) are exclusive events given H. There is no doubt of the mathematical simplifications

that result from this assumption, but we are supposed to be modelling our degrees of belief and it is

questionable whether these have to obey this form of the axiom. Indeed, one of the greatest advocates

of Bayesian theory, Bruno de Finetti, was strongly against the use of P3*. His views can be found in

de Finetti (1972, Section 5.32) or in de Finetti (1974–1975, Section 3.11.3).

There is certainly some arbitrariness about P3*, which is sometimes referred to as an assumption of

σ-additivity, in that it allows additivity over some but not all infinite collections of events

(technically over countable but not over uncountable collections). However, it is impossible in a lot

of contexts to allow additivity over any (arbitrary) collection of events. Thus, if we want a model for

picking a point ‘completely at random’ from the unit interval

it seems reasonable that the probability that the point picked is in any particular sub-interval of the

unit interval should equal the length of that sub-interval. However, this clearly implies that the

probability of picking any one particular x is zero (since any such x belongs to intervals of arbitrarily

small lengths). But the probability that some x is picked is unity, and it is impossible to get one by

adding a lot of zeroes.

Mainly because of its mathematical convenience, we shall assume P3* while being aware of the

problems.

1.1.3 ‘Unconditional’ probability

Strictly speaking, there is, in my view, no such thing as an unconditional probability. However, it

often happens that many probability statements are made conditional on everything that is part of an

individual’s knowledge at a particular time, and when many statements are to be made conditional on

the same event, it makes for cumbersome notation to refer to this same conditioning event every time.

There are also cases where we have so much experimental data in circumstances judged to be

relevant to a particular situation that there is a fairly general agreement as to the probability of an

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