In most real physical experiments we are not, literally, drawing from any ‘urn’. Nevertheless, the idea has turned out to be a useful conceptual device, and in the 250 years since Bernoulli’s Ars Conjectandi it has appeared to scientists that many physical measurements are very much like ‘drawing from Nature’s urn’. But to some the word ‘urn’ has gruesome connotations, and in much of the literature one finds such expressions as ‘drawing from a population’.

在大多数真实的物理实验中，从字面上讲，我们并不是从某个“盒子”中进行采样。尽管如此，这个想法已经证明是一个有用的概念上的手段，并在伯努利的Ars Conjectandi之后的250年中,它向科学家们展示了许多物理测量非常像'从盒中采样'。但对某些人来说，“骨灰盒”这个词很吓人，于是在大多数文献中常常看到这样的表达方式，即“从一群…​中采样”'。

In a few cases, such as recording counts from a radioactive source, survey sampling, and industrial quality control testing, one is quite literally drawing from a real, finite population, and the urn analogy is particularly apt. Then the probability distributions just found, and their limiting forms and generalizations noted in Chapter 7, will be appropriate and useful. In some cases, such as agricultural experiments or testing the effectiveness of a new medical procedure, our credulity can be strained to the point where we see a vague resemblance to the urn problem.

在少数情况下，例如对放射源的进行计数，调查抽样和工业质量控制测试，实际上是从一组真实的有限的东西中进行抽取，这和盒子的类比特别贴切。然后才发现概率分布，并且在第7章中提到的其有限形式和泛化,是合适和有益的。在某些情况下，如农业实验或测试新医疗过程的有效性，可能会让我们轻信,这些问题看起来和从盒子中采样是差不多的。

In other cases, such as flipping a coin, making repeated measurements of the temperature and wind velocity, the position of a planet, the weight of a baby, or the price of a commodity, the urn analogy seems so farfetched as to be dangerously misleading. Yet in much of the literature one still uses urn distributions to represent the data probabilities, and tries to justify that choice by visualizing the experiment as drawing from some ‘hypothetical infinite population’ which is entirely a figment of our imagination. Functionally, the main consequence of this is strict independence of successive draws, regardless of all other circumstances. Obviously, this is not sound reasoning, and a price must be paid eventually in erroneous conclusions.

在其他情况下，例如翻转硬币，重复测量温度和风速，行星的位置，婴儿的体重，或商品的价格，这个盒子的类比似乎是如此牵强，以至于具有危险的误导性。然而在很多文献中,仍然使用盒子中的分布来表示数据的相应概率，并尝试通过将实验看作从某些“假设无限的一组…​”进行采样,来证明这样处理是正确的，但这完全是我们虚幻的臆想。从功能上将，这样做的主要结果是假设了连续抽签中彼此的严格独立性，而不管其他所有情况。显然，这显然是不合理的，最终的代价必然是得出错误的结论。

This kind of conceptualizing often leads one to suppose that these distributions represent not just our prior state of knowledge about the data, but the actual long-run variability of the data in such experiments. Clearly, such a belief cannot be justified; anyone who claims to know in advance the long-run results in an experiment that has not been performed is drawing on a vivid imagination, not on any fund of actual knowledge of the phenomenon. Indeed, if that infinite population is only imagined, then it seems that we are free to imagine any population we please.

这种概念化经常导致人们假设这些分布代表的不仅仅是我们对数据的先验知识，而是这些实验数据的真实的长期变化的认知。显然，这种信念是不能被证实的;任何人,声称知道尚未进行的实验的长期结果,都只是来自生动的想象力，而不是任何有关该现象的实际知识。实际上，如果那些无限的被采样的对象集只是想象出来的，那么我们似乎可以自由地想象出任何的采样集。

From a mere act of the imagination we cannot learn anything about the real world. To suppose that the resulting probability assignments have any real physical meaning is just another form of the mind projection fallacy. In practice, this diverts our attention to irrelevancies and away from the things that really matter (such as information about the real world that is not expressible in terms of any sampling distribution, or does not fit into the urn picture, but which is nevertheless highly cogent for the inferences we want to make). Usually, the price paid for this folly is missed opportunities; had we recognized that information, more accurate and/or more reliable inferences could have been made.

仅仅是想象力的行为，我们无法了解现实世界。认为得出的概率具有任何真实的物理意义,这只是臆想谬误的又一种形式。在实践中，这将我们的注意力转移到无关紧要的地方，远离真正重要的事情（例如那些用任何抽样分布的术语都都无法表达的真实世界的信息，或者不适用于盒中采样的图景，但是对于我们的推理理论非常有用的）。通常，为这种愚蠢而付出的代价是错失了某些机遇;如果我们认识到了这些信息，我们本可以更准确和/或更可靠的进行可靠的推理。

Urn-type conceptualizing is capable of dealing with only the most primitive kind of information, and really sophisticated applications require us to develop principles that go far beyond the idea of urns. But the situation is quite subtle, because, as we stressed before in connection with G¨odel’s theorem, an erroneous argument does not necessarily lead to a wrong conclusion. In fact, as we shall find in Chapter 9, highly sophisticated calculations sometimes lead us back to urn-type distributions, for purely mathematical reasons that have nothing to do conceptually with urns or populations. The hypergeometric and binomial distributions found in this chapter will continue to reappear, because they have a fundamental mathematical status quite independent of arguments that we used to find them here.2

盒中采样问题的概念化,只能处理最原始类型的信息,而真正复杂的应用程序要求我们制定出的原则远远超出了盒中采样的想法。但情况非常微妙，因为正如我们之前所强调的那样与Godel定理有关，错误的论证并不一定导致错误的结论。事实上，正如我们将在第9章中发现的那样，高度复杂的计算,因为纯粹的数学原因而在概念上与盒中采样或被采样集无关，有时会把我们带回到盒中采样这种类型的分布。本章中讨论的超几何和二项式分布可能重新出现，因为它们具有一些根本的数学状态,完全独立于这里得出这些分布的讨论.[2]

On the other hand, we could imagine a different problem in which we would have full confidence in urn-type reasoning leading to the binomial distribution, although it probably never arises in the real world. If we had a large supply $\{U_1,U_2, ... ,U_n\}$ of urns known to have identical contents, and those contents are known with certainty in advance – and then we used a fresh new urn for each draw – then we would assign P(A) = M/N for every draw, strictly independently of what we know about any other draw. Such prior information would take precedence over any amount of data. If we did not know the contents (M, N) of the urns – but we knew they all had identical contents – this strict independence would be lost, because then every draw from one urn would tell us something about the contents of the other urns, although it does not physically influence them.

另一方面，我们可以想出另一个的不同问题,我们有充分的信心在这个问题中用盒中采样的方法得出二项式分布,虽然这可能永远不会出现在现实世界中。如果我们有非常多的盒子$\{U_1，U_2，...，U_n\}$,已知它们都具有相同的内容，并且这些内容事先已经确定--然后每次抽取都换成从一个新的盒子中取--那么对每次抽取赋值为P（A）= M / N,严格独立于我们对任何其他次采样的了解。这样的先验信息将优先于任何其他的信息或数据。如果我们不知道盒子内容的信息（M，N）-- 但我们知道它们的内容都一样 - 这将会失去种严格的独立性，因为那样每次的采样都会告诉我们一些关于盒子的内容的信息，虽然在物理上对这些盒子没有任何影响。

From this we see once again that logical dependence is in general very different from causal physical dependence. We belabor this point so much because it is not recognized at all in most expositions of probability theory, and this has led to errors, as is suggested by Exercise 3.6. In Chapter 4 we shall see a more serious error of this kind (see the discussion following Eq. (4.29)). But even when one manages to avoid actual error, to restrict probability theory to problems of physical causation is to lose its most important applications. The extent of this restriction – and the magnitude of the missed opportunity – does not seem to be realized by those who are victims of this fallacy.

由此我们再次看到，逻辑依赖性通常与物理因果关系非常不同。我们非常重视这一点，因为它在概率论的大多数论述中都没有得到承认，并导致了错误，正如练习3.6所暗示的那样。在第4章中，我们将看到这种类型的更严重的错误（见等式（4.29）之后的讨论）。但即使有人设法避免实际错误，将概率论限制在物理因果关系的问题上,将失去概率论最重要的应用。这种限制的程度 - 以及错失机会的程度 - 错误的受害者们似乎并没有意识到这一点。

Indeed, most of the problems we have solved in this chapter are not considered to be within the scope of probability theory, and do not appear at all in those expositions which regard probability as a physical phenomenon. Such a viewre stricts one to a small subclass of the problems which can be dealt with usefully by probability theory as logic. For example, in the ‘physical probability’ theory it is not even considered legitimate to speak of the probability for an outcome at a specified trial; yet that is exactly the kind of thing about which it is necessary to reason in conducting scientific inference. The calculations of this chapter have illustrated this many times.

实际上，我们在本章中解决的大多数问题都不被认为是在概率论的范围内，并没有出现在那些将概率视为一种物理现象的论述中。这样看的人是将自己限制在,将概率论视为一种逻辑的方法能够处理的问题的一个小的子类中。例如，在“物理概率”理论中，甚至认为说一次实验的结果的概率是什么,这种说法都是不合法的;然而，这正是有必要用科学推断的方法来进行推理的情况。本章中的计算已经多次说明了这一点。

In summary: in each of the applications to follow, one must consider whether the experiment is really ‘like’ drawing from an urn; if it is not, then we must go back to first principles and apply the basic product and sum rules in the new context. This may or may not yield the urn distributions.

总结：在接下来的每种应用中，我们必须考虑实验是否真的"类似"从盒子里进行抽取;如果不是，那么我们必须回到第一原则,并在新的上下文中下应用基本的加法和乘法规则。这可能会也可能不会得到盒采样的分布。 donehere === 3.11.1 A look ahead 前瞻

The probability distributions found in this chapter are called sampling distributions, or direct probabilities, which indicate that they are of the following form: Given some hypothesis H about the phenomenon being observed (in the case just studied, the contents (M, N) of the urn), what is the probability that we shall obtain some specified data D (in this case, some sequence of red and white balls)? Historically, the term ‘direct probability’ has long had the additional connotation of reasoning from a supposed physical cause to an observable effect. But we have seen that not all sampling distributions can be so interpreted. In the present work we shall not use this term, but use ‘sampling distribution’ in the general sense of reasoning from some specified hypothesis to potentially observable data, whether the link between hypothesis and data is logical or causal.

本章中发现的概率分布称为抽样分布或直接分布 概率，表明它们具有以下形式：给出一些假设H. 关于被观察的现象（在刚研究的情况下，内容（M，N） urn），我们获得一些指定数据D的概率是多少（在这种情况下，有些数据） 红色和白色球的序列）？从历史上看，“直接概率”一词长期存在 从假定的物理原因到可观察的效果的推理的其他内涵。 但我们已经看到并非所有的采样分布都可以如此解释。在现在 工作我们不会使用这个术语，而是使用一般意义上的“抽样分布” 从一些特定的假设推断到潜在的可观察数据，无论是链接 假设与数据之间存在逻辑或因果关系。

Sampling distributions make predictions, such as the hypergeometric distribution (3.22), about potential observations (for example, the possible values and relative probabilities of different values of r ). If the correct hypothesis is indeed known, then we expect the predictions to agree closely with the observations. If our hypothesis is not correct, they may be very different; then the nature of the discrepancy gives us a clue toward finding a better hypothesis. This is, very broadly stated, the basis for scientific inference. Just how wide the disagreement between prediction and observation must be in order to justify our rejecting the present hypothesis and seeking a new one, is the subject of significance tests. It was the need for such tests in astronomy that led Laplace and Gauss to study probability theory in the 18th and 19th centuries.

采样分布进行预测，例如超几何分布（3.22）， 关于潜在的观察（例如，可能的值和相对概率 不同的r）值。如果确实知道了正确的假设，那么我们期望得到 预测与观察结果密切一致。如果我们的假设不正确，他们可能会 非常不同;然后，这种差异的性质为我们提供了寻找更好的线索 假设。这是非常广泛的说明，是科学推理的基础。究竟有多宽 预测和观察之间的分歧必须是为了证明我们的拒绝 目前的假设和寻求新的假设，是重要性检验的主题。是的 需要在天文学中进行这样的测试，这导致拉普拉斯和高斯研究概率论 18世纪和19世纪。

Although sampling theory plays a dominant role in conventional pedagogy, in the real world such problems are an almost negligible minority. In virtually all real problems of scientific inference we are in just the opposite situation; the data D are known but the correct hypothesis H is not. Then the problem facing the scientist is of the inverse type: Given the data D, what is the probability that some specified hypothesis H is true? Exercise 3.3 above was a simple introduction to this kind of problem. Indeed, the scientist’s motivation for collecting data is usually to enable him to learn something about the phenomenon in this way.

虽然抽样理论在传统教育学中占主导地位，但在实际教学中却是如此 世界上这样的问题几乎可以忽略不计。几乎所有真正的问题 科学推断我们处于相反的情况;数据D是已知但正确的 假设H不是。然后科学家面临的问题是反向类型：给定 数据D，某些指定假设H的真实概率是多少？练习3.3 以上是对这类问题的简单介绍。的确，科学家的动机 收集数据通常是为了让他能够了解有关这种现象的一些信息 这条路。

Therefore, in the present work our attention will be directed almost exclusively to the methods for solving the inverse problem. This does not mean that we do not calculate sampling distributions; we need to do this constantly and it may be a major part of our computational job. But it does mean that for us the finding of a sampling distribution is almost never an end in itself.

因此，在目前的工作中，我们的注意力将几乎完全针对 解决逆问题的方法。这并不意味着我们不计算 抽样分布;我们需要不断地做到这一点，它可能是我们的主要部分 计算工作。但它确实意味着对我们来说，采样分布的发现是 本身几乎从未结束。

Although the basic rules of probability theory solve such inverse problems just as readily as sampling problems, they have appeared quite different conceptually to many writers. A new feature seems present, because it is obvious that the question: ‘What do you know about the hypothesis H after seeing the data D?’ cannot have any defensible answer unless we take into account: ‘What did you know about H before seeing D?’ But this matter of previous knowledge did not figure in any of our sampling theory calculations. When we asked: ‘What do you know about the data given the contents (M, N) of the urn?’ we did not seem to consider: ‘What did you know about the data before you knew (M, N)?’

虽然概率论的基本规则同样容易解决这些逆问题 作为抽样问题，它们在概念上与许多作家看起来截然不同。 似乎有一个新功能，因为很明显这个问题：'你知道什么 关于假设H看到数据D？'之后不能有任何可辩护的答案，除非 我们考虑到：'在看到D之前你对H有什么了解？'但是这个问题 以前的知识没有在我们的任何抽样理论计算中得出。什么时候我们 问道：'你知道关于骨灰盒内容（M，N）的数据吗？'我们做了 似乎没有考虑：'在你知道（M，N）之前你对数据了解多少？

This apparent dissymmetry, it will turn out, is more apparent than real; it arises mostly from some habits of notation that we have slipped into, which obscure the basic unity of all inference. But we shall need to understand this very well before we can use probability theory effectively for hypothesis tests and their special cases, significance tests. In the next chapter we turn to this problem.

事实证明，这种明显的不对称性比实际更明显;它主要出现 从我们已经陷入的一些符号习惯中，这些习惯模糊了基本的统一性 所有的推论。但在我们使用概率之前，我们需要很好地理解这一点 理论有效地用于假设检验及其特殊情况，意义检验。下一个 我们转向这个问题。

2 In a similar way, exponential functions appear in all parts of analysis because of their fundamental mathematical properties, although their conceptual basis varies widely.

2 以类似的方式，指数函数出现在分析的所有部分，因为它们具有基本的数学特性，尽管它们的概念基础差异很大。