陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「What are the odds, II: the Venezuelan presidential election」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

In a previous blog post , I discussed how, from a Bayesian perspective , learning about some new information can update one’s perceived odds about how likely an “alternative hypothesis” is, compared to a “null hypothesis” . The mathematical formula here is

At a qualitative level, the Bayesian identity (1) is telling us the following: if an alternative hypothesis was already somewhat plausible (so that the prior odds was not vanishingly small), and the observed event was significantly more likely to occur under hypothesis than under , then the hypothesis becomes significantly more plausible (in that the posterior odds become quite elevated). This is quite intuitive, but as discussed in the previous post, a lot hinges on how one

已知结果和反例

In the previous blog post , this calculation was initially illustrated with the following choices of , , and (thus fulfilling ingredient (i)):

在这类讨论里, I would like to run the same analysis on a numerical anomaly in the recent Venezuelan presidential election of June 28, 2024. Here are the officially reported vote totals for the two main candidates, incumbent president Nicolás Maduro and opposition candidate Edmundo González, in the election:

证明或构造的主线

Let us try to apply the above Bayesian framework to this situation, bearing in mind the caveats that this analysis is only strong as the inputs supplied and assumptions made (for instance, to simplify the discussion, 下面会 not also discuss information from exit polling , which in this case gave significantly different predictions from the percentages above).

The first step (ingredient (i)) is to formulate the null hypothesis , the alternative hypothesis , and the event . Here is one possible choice:

阅读时建议盯住的点

Ingredient (ii) – the prior odds that is true over – is highly subjective, and an individual’s estimation of (ii) would likely depend on, or at least be correlated with, their opinion of the current Venezulan administration. Discussion of this ingredient is therefore more political than mathematical, and I will not attempt to quantify it further here. Now we turn to (iii), the estimation of the probability that occurs given the hypothesis . This cannot be computed exactly wit

One can contrast this analysis with that of the Phillipine lottery in the original post. In both cases the probability of the observed event under the null hypothesis was extremely small. However, in the case of the Venezuelan election, there is a plausible causal chain that leads to an elevated probability of the observed event under the alternative hypothesis, whereas in the case of the lottery, only extremely implausible chains could be constructed that would lead to the s

值得单独记下的条目

• (i) A precise formulation of the null hypothesis and the alternative hypothesis , and the new information ;
• (ii) A reasonable estimate of the prior odds of the alternative hypothesis being true (compared to the null hypothesis );
• (iii) A reasonable estimate of the probability that the event would occur under the null hypothesis ; and
• (iv) A reasonable estimate of the probability that the event would occur under the alternative hypothesis ,
• was the event that the October 1, 2022 PSCO Grand Lotto in the Phillippines drew the numbers (that is to say, consecutive multiples if ), though not necessarily in that order;
• was the null hypothesis that the lottery was fair and the numbers were drawn uniformly at random (without replacement) from the set ; and
• was the alternative hypothesis that the lottery was rigged by some corrupt officials for their personal gain.
• González: 4,445,978 votes

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源