概率论沉思录

• tinylambda

  2019-08-28
  为了关注于具有建设性的事物，远离有争议的不相关的事务，我们应该创造一个假象的存在。它的大脑是由我们设计的，所以其推理是根据明确确定的规则进行的。这些规则是从简单的必须品中被演绎出来的，在我们看来，这些必须品在人类大脑中是可获取的；比如，我们认为一个理性的人，一旦发现他们与其中一个必需品相违背，他们就会修正其想法。原则上，我们可以自由地应用我们喜欢的规则；这是定义我们应该研究哪个机器人的方法。将它的推理同你自己的推理进行比较，如果你发现没有相似之处，你就可以拒绝我们的机器人，并根据你的喜好来设计一个不同的机器人。如果你发现其中有非常强的相似之处，并且想要信任这个机器人，将其用于解决自己的推理问题，那么这将是理论的一个成就，而不是一个前提。我们的机器人会对命题进行推理。正如我们之前看到的，我们将使用斜体大些字母来表示各种命题，{A, B, C, 等等}，目前，我们必须要求所使用的任何命题对机器人来说必须具有明确的含义，同时必须是非真即假的确定性逻辑类型。也就是说，除非另作说明，我们只关心二值逻辑，或者叫做亚里士多德逻辑。我们并不要求这样的‘亚里士多德逻辑’通过可行的调查都能确定其真或者假；事实上，我们做不到这一点通常就是我们需要机器人帮助的原因。比如，作者个人认为如下的两个命题为真：A º 贝多芬和柏辽兹从未见过面B º 贝多芬的音乐比柏辽兹的音乐的质量更加有持续性，尽管鼎盛时期的柏辽兹可以匹敌任何人。命题B是不允许机器人思考的，而命题A是，尽管其真假今天也不太可能确定了。在我们的理论形成之后，我们将有兴趣探究当前对于类似于A的亚里士多德逻辑的限制是否可以放宽，这样机器人就可以帮助我们处理类似于B的更模糊的命题了（见第18章 Ap-分布）。
• tinylambda

  2019-08-28
  模型有完全不同类型的用途。许多人喜欢说，“他们永远造不出一台机器来替代人类思维——这里面包含了一些机器永远都做不到的东西”。J. 冯˙诺伊曼在普林斯顿的一次关于计算机的演讲中给出了一个很好的答案，作者有幸参加。在回答听众的典型问题的时候（‘但是，当然，一台机器不能真正思考，是吗？’），他说到：你坚信有一些事情是机器所不能做的。如果你能够精确地告诉我机器不能做什么，那么，我总是能够制造出做这件事的机器。原则上，机器不能为我们做的操作仅限于那些我们无法详细描述的操作，或者在有限的步骤中无法完成的操作。当然，有些人会联想到哥德尔不完备定理，测不准原理，永不停止的图灵机等等。但要回答所有这些疑问，我们只需要指出人类大脑的存在即可，它确实是存在的。正如冯诺伊曼所指出的，制造“思考机器”的唯一真正限制是我们自己就不知道“思考”到底由什么组成。但是在我们对常识的研究中，我们将得到一些关于思维机制的非常明确的观点。每当我们可以通过指定一组确定的操作来建立一个数学模型以再现一部分常识的时候，就向我们展示了如何“建立一台机器”（即编写一个计算机程序），它对不完整的信息进行操作，通过使用前面的弱三段论的量化版本，来进行合理推理而不是演绎推理。事实上，针对于这种特定推论问题的计算机软件的开发是这个领域中最活跃和有用的当代潮流。这样处理的一种问题可能是：给定大量数据，包括10000个独立的观察，根据这些数据和现有的先验信息，确定100个关于工作原因的不同假设的相对合理性。我们的无意识常识，对于结果非常不同的两个假设，可能足以对其做出决定；但是，要处理100个没有明显不同的假设时，如果没有计算机和一个良好的数学理论来告诉我们如何编程，我们将束手无策。也就是说，在我们的警察三段论（1.5）中，什么决定了A的合理性会大幅度地增加，以近乎可以确定；或者仅仅是一个可以忽略的幅度，让数据B变得几乎不相...
• find_my_way

  2017-09-22
  The fundamental, inescapable distinction between probability and frequency lies in this relativity principle: probabilities change when we change our state of knowledge; frequencies do not.
• find_my_way

  2017-08-31
  We have to recognize that our robot is immature; it reasons like a four-year-old child does. The remarkable thing about small children is that you can tell them the most ridiculous things and they will accept it all with wide open eyes, open mouth, and it never occurs to them to question you. They will believe anything you tell them.
• find_my_way

  2017-08-04
  One might expect that open discussion of public issues would tend to bring about a general consensus. On the contrary, we observe repeatedly that when some controversial issue has been discussed vigorously for a few years, society becomes polarized into two opposite extreme camps; it is almost impossible to find anyone who retains a moderate view. Prob- ability theory as logic shows how two persons, given the same information, may have their opinions driven in opposite directions by it, and what must be done to avoid this.
• ywis

  2014-03-24
  It is the conceptual matters: how to make the initial connection between the real-world problem and the abstract mathematics.
• [已注销]

  2012-07-06
  第一章介绍了plausible reasoning的概念，plausible reasoning不是严格的推理，二是基于可能性的推理。例如如果有A->B，那么~A->~B就不一定严格成立，而是有可能。我们在思考问题的时候经常会基于自己的历史经验使用plausible reasoning。同时，作者给出了数理逻辑中的布尔代数、布尔代数的运算、用与或非表示所有真值函数的范式、NAND/NOR的介绍。最后，作者给出了关于plausible reasoning的三条基本假设：1，Representation of degrees of plausibility by real numbers2，Qualitative correspondence with common sense3，Consistency作者将在这三条假设的基础上推导自动化的inference的定理，作者把它称为thinking robot。There is only one set of mathematical operations for manipulating plausibilities which has all these properties.
• [已注销]

  2012-07-06
  George P´olya on ‘Mathematics and Plausible Reasoning’. He dissected our intuitive ‘common sense’ into a set of elementary qualitative desiderata.In the writer’s lectures, the emphasis was therefore on the quantitative formulation ofP´olya’s viewpoint, so it could be used for general problems of scientific inference, almost all of which arise out of incomplete information rather than ‘randomness’按照作者的意思，是在Polya工作的基础上，对基于incomplete information的scientific inference作一个量化的扩展。For many years, there has been controversy over ‘frequentist’ versus ‘Bayesian’ methods of inference, in which the writer has been an outspoken partisan on the Bayesian side.However, neither the Bayesian nor the frequentist approach is universally applicable, so in the present, more general, work we take a broader ...
• 和大人

  2011-05-04
  Many people are fond of saying, ‘They will never make a machine to replace the human mind – it does many things which no machine could ever do.’ A beautiful answer to this was given by J. von Neumann in a talk on computers given in Princeton in 1948, which the writer was privileged to attend. In reply to the canonical question from the audience (‘But of course, a mere machine can’t really think, can it?’), he said:You insist that there is something a machine cannot do. If you will tell me precisely what it is that a machine cannot do, then I can always make a machine which will do just that!
• 和大人

  2011-05-03
  Of course, on publishing a new theorem, the mathematician will try very hard to invent an argument which uses only the first kind; but the reasoning process which led to the theorem in the first place almost always involves one of the weaker forms (based, for example, on following up conjectures suggested by analogies). The same idea is expressed in a remark of S. Banach (quoted by S. Ulam, 1957):Good mathematicians see analogies between theorems; great mathematicians see analogies between analogies.
• Once

  2011-11-24
  How could we build a machine which would carry out useful plausible reasoning, following clearly defined priciples expressing an idealized common sense?
• Once

  2011-11-24
  How can we build a mathematical model of human common sense?
• find_my_way

  2017-07-26
  Good mathematicians see analogies between theorems; great mathematicians see analogies between analogies.
• masterplan

  2018-08-30
  The writer has learned from much experience that this primary emphasis on the logic of the problem, rather than the mathematics, is necessary in the early stages. For modern students, the mathematics is the easy part; once a problem has been reduced to a definite mathematical exercise, most students can solve it effortlessly and extend it endlessly, without further help from any book or teacher. It is in the conceptual matters (how to make the initial connection between the real-world problem and the abstract mathematics) that they are perplexed and unsure how to proceed.